\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 258, pp. 1--29.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/258\hfil Product measurability]
{Product measurability with applications to a stochastic contact
problem with friction}

\author[K. L. Kuttler, M. Shillor \hfil EJDE-2014/258\hfilneg]
{Kenneth L. Kuttler, Meir Shillor} % in alphabetical order

\address{Kenneth L. Kuttler \newline
 Department of Mathematics,
Brigham Young University,
Provo, UT 84602, USA}
\email{klkuttle@math.byu.edu}

\address{Meir Shillor \newline
Department of Mathematics and Statistics,
Oakland University,
Rochester, MI 48309, USA}
\email{shillor@oakland.edu}

\thanks{Submitted August 27, 2014. Published December 11, 2014.}
\subjclass[2000]{60H15, 34F05, 35R60, 60H10, 74M10}
\keywords{Stochastic differential equations; product measurability;
\hfill\break\indent dynamic contact}

\begin{abstract}
 A new product measurability result for evolution equations with random inputs,
 when there is no uniqueness of the $\omega$-wise problem, is established
 using results on measurable
 selection theorems for measurable multi-functions. The abstract result is
 applied to a general stochastic system of ODEs with delays and to a
 frictional contact problem in which the gap between a
 viscoelastic body and the foundation and the motion of the foundation are
 random processes. The existence and uniqueness of a measurable solution
 for the problem with Lipschitz friction coefficient, and just existence for
 a discontinuous one, is obtained by using a sequence of approximate
 problems and then passing to the limit. The new result shows that the limit
 exists and is measurable. This new result opens the way to establish the
 existence of measurable solutions for various problems with random
 inputs in which the uniqueness of the solution is not known, which is 
 the case in many problems involving frictional contact.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction}
\label{sec:intro}

This article establishes the product measurability of solutions to
evolution equations having random coefficients, that is, the
various operators occurring in the equations are assumed to be
stochastic processes depending on the random variable $\omega $
that belongs to a probability space $(\Omega,\mathcal{F},P)$.
In many problems described by nonlinear
partial differential equations and inclusions, this is an important
generalization. We apply our theory to a system of ordinary
delay-differential equations involving inputs that are stochastic
processes and to a problem of frictional contact between a viscoelastic
body and a reactive foundation. In the latter problem the gap between the
body and the foundation in the reference configuration is assumed to be
a random process, and so is the speed of the foundation.

This abstract result, Theorem 1, opens the way to study a host of models set
as differential inclusions or equations that arise in many applications in
which some of the input parameters are naturally random or known with some
uncertainty, which is the case in most applied continuous systems. This
general result on the measurability of the solution is based on the use of
theorems on measurable multi-functions. This approach allows one to
essentially consider the problem for one fixed value of the random variable
$\omega $ and then to conclude the existence of a measurable solution. A
related contact problem for the vibrations of a Gao beam when the gap is
random has been studied in \cite{KLS14}, however since the setting there was
simpler, the result was obtained directly. This abstract result applies
directly to the stochastic Navier-Stokes equations studied in \cite{ben73}.

From the applied point of view, it is natural to allow various parameters in
the problem to be random variables or stochastic processes. However,
such generalizations lead to the difficulty of showing that the solutions
are measurable, although showing measurability is relatively
straightforward when the $\omega$-wise problems obtained by fixing $\omega $
has a unique solution. Our approach is more general and it
does not require the uniqueness of the solution of the
problem with fixed $\omega $. In addition, we do not make any special
assumptions on the underlying probability space $(\Omega ,\mathcal{F},P)$,
in contrast to what was done in the important paper \cite{ben73} in the case
of the Navier-Stokes equations. However, we use the same measurable
selection theorem for measurable multi-functions but in a very different
context.

The main application of Theorem 1 in this work is to a stochastic version of
a model for the dynamic frictional contact between a viscoelastic body and a
reactive foundation when the coefficient of friction is slip-rate dependent,
that was studied in \cite{KS02}. The problem without stochastic input but
with reactive foundation and slip-rate independent friction coefficient was
first studied in \cite{Mar87} and then in \cite{KMS88, KMS89} where the
static case was considered, and since then in many papers with models of
various degrees of complexity, see, e.g., \cite{and97, FT95, kut97, KS01-2,
KS04-1, KS04-3, RSS98-1} and the references therein. General references
about various versions of related contact problems with friction are, e.g.
\cite{duv76, EJK05, HS01, MOS13, nan95, SST04book} and the references
therein. In addition to adding stochastic inputs, we also present an
improved result for the case when the friction coefficient is a
discontinuous function of the slip-rate or even a graph, than in our earlier
papers \cite{KS01-2, KS04-3}. These methods open the way to study a variety
of contact problems in which the various parameters and inputs are random.
We foresee that it will be used in a number of publications.

Following the Introduction, the main theorem, Theorem 1, is formulated and
proved in Section 2. It provides a general approach that allows one to use
standard techniques for evolution equations and inclusions for fixed value
of the random variable $\omega$. The main constraints are that one must
start with measurable functions and that subsequences converge weakly to
weakly continuous functions, which is usually the case in evolution
problems. An application of the theorem to the measurability of the
solutions of systems of ordinary differential equations or inclusions with
delays, when some of the inputs are random, is provided in Section 3.
The contact problem with random gap and sliding rate is studied in Section
4, where the problem data is given, too. Subsection 4.1 provides an abstract
form of the problem, and contains some results from the literature on
compact sets in function spaces. To deal with the friction term, which is a
set-inclusion, the problem is regularized in Subsection 4.2 and the Galerkin
method is used to obtain approximate measurable solutions. Then, by
obtaining the necessary a priori estimates, we pass to the limit and obtain
the unique measurable solution of the problem in the case when the
coefficient of friction is a Lipschitz function of the slip-rate. Finally,
in Subsection 4.3 the case when the friction coefficient is discontinuous,
has a jump from a static value to a dynamic value when relative motion
commences, is studied. In this case the uniqueness of the solution is not
known, and seems to be unlikely. Using the new tools, we establish
the existence of a measurable solution to the problem. As noted above, 
this is an improvement of the result in \cite{KS02}.

\section{The measurable selection theorem}
\label{sec:select}

In this section we study the problem of obtaining product measurable
solutions to evolution equations in the context when either there is no
uniqueness to the non-stochastic problem obtained by fixing a given $\omega $
in the probability space, or uniqueness is not known. This is of
considerable interest because there are many important problems in which the
existence of solutions is known but not their uniqueness. This often occurs
when weak limits are used to obtain existence but there is insufficient
monotonicity to show uniqueness. For example, the equations describing a
vibrating purely elastic Gao beam appear to fail to have uniqueness, see
\cite{KLS14}. Another well known example is the three-dimensional
Navier-Stokes equations for an incompressible viscous fluid with Dirichlet
boundary conditions in a bounded domain.

We make essential use of the ideas of measurable multi-functions having
values in a complete separable metric space, i.e., a Polish space,
 \cite[Vol. 1, p.141]{pap97}.

\begin{definition}\rm
Let $X$ be a Polish space and let $(\Omega ,\mathcal{F})$ be a
measurable space and let $F:\Omega \to 2^{X}$ be a multi-function
assumed to have values that are non-empty and closed sets. Then, $F$ is
said to be \emph{measurable} if for every open set $U$ in $2^{X}$,
\[
F^{-}(U) =\{ \omega :F(\omega ) \cap U\neq
\emptyset \} \in \mathcal{F}.
\]
The multi-function is said to be {\rm strongly measurable} if for every
closed set $C$ in $2^{X}$,
\[
F^{-}(C) =\{ \omega :F(\omega ) \cap C\neq
\emptyset \} \in \mathcal{F}.
\]
\end{definition}

One can show that strong measurability implies measurability and that
measurability is sufficient to obtain the existence of a measurable
selection, which is a function $\gamma (\omega )$ that is
$\mathcal{F}$ measurable and $\gamma (\omega ) \in F(\omega
)$ for each $\omega $. In the case when the values of $F$ are
compact sets, it can be shown that the two versions of measurability are
equivalent (the proof can be found in \cite[Vol. 1, p.143]{pap97}).

We now introduce some notation. We describe randomness by a probability space
$(\Omega ,\mathcal{F},P)$, where $\Omega $ is the \textit{sample space} with
elements $\omega $, $\mathcal{F}$ is a given $\sigma $-algebra of subsets of
$\Omega $, and $P$ is the \textit{\ probability measure} on $\mathcal{F}$.
The usual Borel $\sigma $-algebra of open sets in $[0,T]$ is denoted
by $\mathcal{B}([ 0,T] )$ and $P=\mu _{L}$ is the usual
Lebesgue measure. Next, $C=C(\alpha ,\dots ,\beta )$ denotes a positive
constant that depends only on the problem data and on $\alpha ,\dots ,\beta $
, whose value may change from place to place. Also, $C^{0,1}([ 0,T
] )$ denotes the H\"older space with $\gamma =1$, so that
the norm is
\[
\| f\| _{0,1}=\sup_{t\in [ 0,T] }|
f(t) | +\sup \big\{ \frac{| f(t) -f(s) | }{| t-s| }:s\neq t\big\} .
\]

The following abstract theorem is the main result in this work.

\begin{theorem}\label{thm1}
Let $V$ be a reflexive separable Banach space with dual $
V'$, and let $p,p'$ be such that $p>1$ and $\frac{1}{p}+
\frac{1}{p'}=1$, and $\omega\in \Omega$. Let the functions
$t\to u_n(t,\omega )$, for $n\in \mathbb{N}$, be in
 $L^{p'}( [ 0,T] ;V')$ and $(t,\omega )
\to u_n(t,\omega )$ be
$\mathcal{B}([ 0,T] ) \times \mathcal{F}\equiv \mathcal{P}$ measurable into
$V'$. Suppose there is a set of measure zero $N\subseteq \Omega $
such that if $\omega \notin N$, then
\begin{equation}
\sup_{t\in [ 0,T] }\| u_n(t,\omega )
\| _{V'}\leq C(\omega ) , \label{21a}
\end{equation}
for all $n$. Also, suppose for each $\omega \notin N$, each subsequence
of $\{ u_n\} $ has a further subsequence that converges weakly
in $L^{p'}([ 0,T] ;V')$ to $v(
\cdot ,\omega ) \in L^{p'}([ 0,T] ;V')$ such that the function $t\to v(t,\omega )$
is weakly continuous into $V'$.

Then, there exists a product measurable function $u$ such that $t\to
u(t,\omega )$is weakly continuous into $V'$ for
each $\omega \notin N$. Moreover, there exists, for each $\omega \notin
N, $ a subsequence $u_{n(\omega ) }$ such that $u_{n(
\omega ) }(\cdot ,\omega ) \to u(\cdot
,\omega )$ weakly in $L^{p'}([ 0,T]
;V')$.
\end{theorem}

We prove the theorem in steps given below. We let $X=\prod_{k=1}^{\infty
}C([ 0,T] )$ and note that when it is equipped with
the product topology, then one can consider $X$ as a metric space using the
metric
\[
d(\mathbf{f,g}) \equiv \sum_{k=1}^{\infty }2^{-k}\frac{
\| f_k-g_k\| }{1+\| f_k-g_k\| },
\]
where $\mathbf{f}=(f_1, f_2, \dots)$, $\mathbf{g}=(g_1, g_2, \dots)
\in X$, and the norm is the maximum norm in $C([ 0,T]
)$. With this metric, $X$ is complete and separable.

The next lemma claims that if $\{ \mathbf{f}_n\} $ has each
component bounded in $C^{0,1}([ 0,T] )$ then it is
pre-compact in $X$.

\begin{lemma}\label{lemma2}
Let $\{ \mathbf{f}_n\} $ be a sequence in $X$
and suppose that each one of the components $f_{nk}$ is bounded in $
C^{0,1}([ 0,T] )$ by $C=C(k)$. Then, there exists a
subsequence $\{ \mathbf{f}_{n_j}\} $ that converges to some $
\mathbf{f}\in X$ as $n_j\to \infty$. Thus, $\{ \mathbf{f}_n\}
$ is pre-compact in $X$.
\end{lemma}

\begin{proof}
By the Ascoli-Arzela theorem, there exists a subsequence $\{ \mathbf{f}
_{n_1}\} $ such that the sequence of the first components $
f_{n_11} $ converges in $C([ 0,T] )$. Then, taking
a subsequence, one can obtain $\{n_2\}$ a subsequence of $\{n_1\}$ such
that both the first and second components of $\mathbf{f} _{n_2}$ converge.
Continuing in this way one obtains a sequence of subsequences, each a
subsequence of the previous one such that $\mathbf{f} _{n_j}$ has the
first $j$ components converging to functions in $C([ 0,T]
)$. Therefore, the diagonal subsequence has the property that it has
every component converging to a function in $C([ 0,T]
)$. The resulting function is $\mathbf{f}\in \prod_kC([
0,T] )$.
\end{proof}


Now, for $m\in \mathbb{N}$ and $\phi \in V'$, define
$l_m(t)\equiv \max (0,t-(1/m) )$ and $\psi
_{m,\phi }:L^{p'}([ 0,T] ;V')
\to C([ 0,T ] )$ by
\[
\psi _{m,\phi }u(t) \equiv \int_0^{T}\langle m\phi
\mathcal{X} _{[ l_m(t),t] }(s) ,u(s)
\rangle _Vds=m\int_{l_m(t)}^{t}\langle \phi ,u(s)
\rangle _Vds.
\]
Here, $\mathcal{X}_{[ l_m(t),t] }(\cdot)$ is the
characteristic function of the interval $[ l_m(t),t]$ and $\langle
\cdot, \cdot \rangle _V$ is the duality pairing between $V$
and $V'$.

Let $\mathcal{D}=\{ \phi _r\} _{r=1}^{\infty }$ denote a
countable dense subset of $V$. Then, the pairs $(m,\phi ) \in
\mathbb{N}\times \mathcal{D}$ form a countable set, and let
$(m_k,\phi _{r_k})$ denote an enumeration of these pairs. To
simplify the notation, we set
\[
f_k(u) (t) \equiv \psi _{m_k,\phi
_{r_k}}(u) (t)
=m_k\int_{l_{m_k}(t)}^{t}\langle \phi _{r_k},u(s)
\rangle _Vds.
\]

For fixed $\omega \notin N$ and $k$, the functions $\{ t\to
f_k(u_j(\cdot ,\omega ) ) (t)
\} _j$ are uniformly bounded and equicontinuous because they
are in $C^{0,1}([ 0,T] )$. Indeed, we have
\[
| f_k(u_j(\cdot ,\omega ) ) (
t) | =\big| m_k\int_{l_{m_k}(t)}^{t}\langle
\phi _{r_k},u_j(s,\omega ) \rangle _Vds \big|
\leq C(\omega ) \| \phi _{r_k}\| _V,
\]
and for $t\leq t'$,
\begin{align*}
&| f_k(u_j(\cdot ,\omega ) ) (
t) -f_k(u_j(\cdot ,\omega ) ) (t') | \\
&\leq \Big| m_k\int_{l_{m_k}(t)}^{t}\langle \phi
_{r_k},u_j(s,\omega ) \rangle _Vds
-m_k\int_{l_{m_k}(t')}^{t'}\langle \phi
_{r_k},u_j(s,\omega ) \rangle _V\,ds\Big| \\
&\leq 2m_k| t'-t| C(\omega )
\| \phi _{r_k}\| _{V'}.
\end{align*}
By Lemma \ref{lemma2}, the set of functions $\{ \mathbf{f}(
u_j(\cdot ,\omega ) ) \} _{j=n}^{\infty }$ is
pre-compact in the space $X=\prod_kC([ 0,T] ) $. We now
define a set valued map $\Gamma ^{n}:\Omega \to X$ by
\[
\Gamma ^{n}(\omega ) \equiv \overline{\cup _{j\geq n}\{
\mathbf{f }(u_j(\cdot ,\omega ) ) \} },
\]
where the closure is taken in $X$. Then, $\Gamma ^{n}(\omega )$
is the closure of a pre-compact set in $X$ and so $\Gamma ^{n}(\omega
)$ is compact in $X$. From the definition, a function $\mathbf{f}$ is
in $\Gamma ^{n}(\omega )$ if and only if $d(\mathbf{f,f}
(w_{l}) ) \to 0$ as $l\to \infty $, where
each $w_{l}$ is one of the $u_j(\cdot,\omega )$ for $j\geq n$.
 In the topology on $X$, this happens if and only if for every $k$,
\[
f_k(t) =\lim_{l\to \infty
}m_k\int_{l_{m_k}(t)}^{t}\langle \phi _{r_k},w_{l}(s,\omega
) \rangle _Vds,
\]
where the limit is the uniform limit in $t$.

\begin{lemma}\label{lemma4}
The mapping $\omega \to \Gamma ^{n}(\omega
)$ is an $\mathcal{F}$ measurable set-valued map with values in $X$.
If $\sigma $ is a measurable selection, then for each $t$, $\omega
\to \sigma (t,\omega )$ is $\mathcal{F}$ measurable and
$(t,\omega ) \to $ $\sigma (t,\omega )$ is $
\mathcal{B}([ 0,T] ) \times \mathcal{F}$ measurable.
\end{lemma}

We note that if $\sigma $ is a measurable selection then $\sigma (
\omega ) \in \Gamma ^{n}(\omega )$, so $\sigma =\sigma
(\cdot ,\omega )$ is a continuous function. To have $\sigma$
measurable  means that $\sigma _k^{-1}(\text{open}) \in
\mathcal{F}$, where the open set is in $C([ 0,T] )$.


\begin{proof}[Proof of Lemma \ref{lemma4}]
Let $O$ be a basic open set in $X$ so that $O=\prod_{k=1}^{\infty }O_k$,
where $O_k$ is a proper open set of $C([ 0,T] )$
only for $k\in \{ k_1,\cdots ,k_r\} $, while in the rest of
the components the open set is the whole space $C([ 0,T]
)$. We need to show that
\[
\Gamma ^{n-}(O) \equiv \{ \omega :\Gamma ^{n}(\omega
) \cap O\neq \emptyset \} \in \mathcal{F}.
\]
Now, $\Gamma ^{n-}(O) =\cap _{i=1}^{r}\{ \omega :\Gamma
^{n}(\omega ) _{k_{i}}\cap O_{k_{i}}\neq \emptyset \} $,
so we consider whether
\begin{equation}
\{ \omega :\Gamma ^{n}(\omega ) _{k_{i}}\cap O_{k_{i}}\neq
\emptyset \} \in \mathcal{F}. \label{21}
\end{equation}
From the definition of $\Gamma ^{n}(\omega ) $, this is
equivalent to the condition that $f_{k_{i}}(u_j(\cdot ,\omega
) ) =(\mathbf{f}(u_j(\cdot ,\omega )
) ) _{k_{i}}\in O_{k_{i}}$ for some $j\geq n$, and so the set in
\eqref{21} is of the form
\[
\cup _{j=n}^{\infty }\{ \omega :(\mathbf{f}(u_j(
\cdot ,\omega ) ) ) _{k_{i}}\in O_{k_{i}}\} .
\]
Now $\omega \to (\mathbf{f}(u_j(\cdot ,\omega
) ) ) _{k_{i}}$ is $\mathcal{F}$ measurable into $C(
[ 0,T] )$ and so the above set is in $\mathcal{F}$. To see
this, let $g\in C([ 0,T] )$ and consider the inverse
image of the ball with radius $r$ and center $g$,
\[
B(g,r) =\{ \omega :\| (\mathbf{f}(
u_j(\cdot ,\omega ) ) ) _{k_{i}}-g\|
_{C([ 0,T] ) }<r\} .
\]
By continuity considerations,
\[
\| (\mathbf{f}(u_j(\cdot ,\omega ) )
) _{k_{i}}-g\| _{C([ 0,T] )
}=\sup_{t\in \mathbb{Q}\cap [ 0,T] }| (\mathbf{f}
(u_j(t,\omega ) ) ) _{k_{i}}-g(
t) | ,
\]
which is the supremum over countably many $\mathcal{F}$ measurable functions
and so it is $\mathcal{F}$ measurable. Since every open set is the countable
union of such balls, the $\mathcal{F}$ measurability follows. Hence, $\Gamma
^{n-}(O)$ is $\mathcal{F}$ measurable whenever $O$ is a basic
open set.

Now, $X$ is a separable metric space and so every open set is a countable
union of these basic sets. Let $U\subseteq X$ be open with $U=\cup
_{l=1}^{\infty }O_{l}$ where $O_{l}$ is such a basic open set. Then,
\[
\Gamma ^{n-}(U) =\cup _{l=1}^{\infty }\Gamma ^{n-}(
O_{l}) \in \mathcal{F}.
\]
The existence of a measurable selection follows from the standard theory of
measurable multi-functions \cite{aub90} or \cite[Vol. 1, Page 141]{pap97}.
 If $\sigma $ is one of these measurable
selections, the evaluation at $t$ is $\mathcal{F}$ measurable. Thus, $\omega
\to \sigma (t,\omega )$ is $\mathcal{F}$ measurable
with values in $\mathbb{R}^{\infty }$. Also, $t\to \sigma (
t,\omega )$ is continuous, and so it follows that in fact $\sigma $
is product measurable as claimed.
\end{proof}


\begin{definition} \rm
Let $\Gamma (\omega ) \equiv \cap _{n=1}^{\infty }\Gamma
^{n}(\omega )$.
\end{definition}

\begin{lemma}\label{30decl1f}
$\Gamma $ is a nonempty $\mathcal{F}$ measurable set-valued
function with values in compact subsets of $X$. There exists a measurable
selection $\gamma $ such that $(t,\omega ) \to \gamma
(t,\omega )$ is $\mathcal{P}$ measurable. Also, for each $
\omega $, there exists a subsequence, $u_{n(\omega ) }(
\cdot ,\omega )$ such that for each $k$,
\[
\gamma _k(t,\omega ) =\lim_{n(\omega ) \to
\infty }\mathbf{f}(u_{n(\omega ) }(t,\omega )
) _k=\lim_{n(\omega ) \to \infty
}m_k\int_{l_{m_k}(t)}^{t}\langle \phi _{r_k},u_{n(\omega
) }(s,\omega ) \rangle _Vds.
\]
\end{lemma}

\begin{proof}
From the definition of $\Gamma (\omega ) =\cap _{n=1}^{\infty
}\Gamma ^{n}(\omega )$ it follows that $\omega \to
\Gamma (\omega )$ is a compact set-valued map in $X$ and is
nonempty because each $\Gamma ^{n}(\omega )$ is nonempty and
compact, and the $\Gamma ^{n}(\omega )$ are nested. We next
show that $\omega \to \Gamma (\omega )$ is $\mathcal{F}$
measurable. Indeed, each $\Gamma ^{n}$ is compact valued and $\mathcal{F}$
measurable so, if $F$ is closed,
\[
\Gamma (\omega ) \cap F=\cap _{n=1}^{\infty }\Gamma ^{n}(
\omega ) \cap F,
\]
and the left-hand side is not empty iff each $\Gamma ^{n}(\omega
) \cap F\neq \emptyset $. Thus, for $F$ closed,
\[
\{ \omega :\Gamma (\omega ) \cap F\neq \emptyset \}
=\cap _n\{ \omega :\Gamma ^{n}(\omega ) \cap F\neq
\emptyset \},
\]
and so
\[
\Gamma ^{-}(F) =\cap _n\Gamma ^{n-}(F) \in
\mathcal{F}.
\]
The last claim follows from the theory of multi-functions, see, e.g., \cite{aub90,
pap97}. The fact that $\Gamma ^{n}(\omega )$ is compact
implies that strong measurability and measurability coincide,
\cite[Vol. 1, p.143]{pap97}. Thus, $\Gamma ^{n}$ is measurable and $\Gamma ^{n-}(
U) \in \mathcal{F}$ , for $U$ open, implies $\Gamma ^{n-}(
F) \in \mathcal{F}$ for $F$ closed. Thus, $\omega \to
\Gamma (\omega )$ is nonempty compact valued in $X$ and
strongly $\mathcal{F}$ measurable.

Standard theory, \cite[Vol. 1, pp 141-2]{pap97}, also guarantees the existence of
an $\mathcal{F}$ measurable selection $\omega \to \gamma (\omega
)$ with $\gamma (\omega ) \in $ $\Gamma (\omega
)$, for each $\omega $,
and also that $t\to \gamma _k(t,\omega )$ (the $k$th
component of $\gamma $) is continuous. Next, we consider the product
measurability of $\gamma _k$. We know that $\omega \to \gamma
_k(\omega )$ is $\mathcal{F}$ measurable into $C([
0,T] )$ and since pointwise evaluation is continuous, $\omega
\to \gamma _k(t,\omega )$ is $\mathcal{F}$
measurable. (Indeed, a continuous function of a measurable function is
measurable.) Then, since $t\to \gamma _k(t,\omega )$
is continuous, it follows that $\gamma _k$ is a $\mathcal{P}$ measurable
real valued function and that $\gamma $ is a $\mathcal{P}$ measurable $
\mathbb{R}^{\infty }$ valued function. Since $\gamma (\omega )
\in \Gamma (\omega ) $, it follows that for each $n,\gamma
(\omega ) \in \Gamma ^{n}(\omega ) $. Hence,
there exists $j_n\geq n$ such that for each $\omega $,
\[
d(\mathbf{f}(u_{j_n}(\cdot ,\omega ) )
,\gamma (\omega ) ) <2^{-n}.
\]
Therefore, for a suitable subsequence $\{ u_{n(\omega )
}(\cdot ,\omega ) \} $, we have
\[
\gamma (\omega ) =\lim_{n(\omega ) \to
\infty }\mathbf{f}(u_{n(\omega ) }(\cdot ,\omega
) ) .
\]
for each $\omega $. In particular, for each $k$ and for each $t$, we have
\begin{equation}
\gamma _k(t,\omega ) =\lim_{n(\omega ) \to
\infty }\mathbf{f}(u_{n(\omega ) }(t,\omega )
) _k=\lim_{n(\omega ) \to \infty
}m_k\int_{l_{m_k}(t)}^{t}\langle \phi _{r_k},u_{n(\omega
) }(s,\omega ) \rangle _Vds, \label{22}
\end{equation}
\end{proof}

Note that it is not clear that $(t,\omega ) \to \mathbf{
 f }(u_{n(\omega ) }(t,\omega ) )$
is $\mathcal{P}$ measurable, although $(t,\omega ) \to
\gamma (t,\omega )$ is $\mathcal{P}$ measurable.

We have now all the ingredients needed to prove the theorem.

\begin{proof}[Proof of Theorem \ref{thm1}]
 By assumption, there exists
a subsequence, still denoted by $n(\omega )$, such
that, in addition to \eqref{22}, the weak limit $\lim_{n(\omega
) \to \infty }u_{n(\omega ) }(\cdot ,\omega
) =u(\cdot ,\omega )$ exists in $L^{p'}(
[ 0,T ] ;V')$ such that $t\to u(
t,\omega )$ is weakly continuous into $V'$. Then, \eqref{22}
also holds for this further subsequence and in addition,
\[
m_k\int_{l_{m_k}(t)}^{t}\langle \phi _{r_k},u(s,\omega
) \rangle _Vds=\lim_{n(\omega ) \to \infty
}m_k\int_{l_{m_k}(t)}^{t}\langle \phi _{r_k},u_{n(\omega
) }(s,\omega ) \rangle _Vds=\gamma _k(
t,\omega ).
\]
Let $\phi \in \mathcal{D}$ be given, then there exists a subsequence,
denoted by $k$, such that $m_k\to \infty $ and $\phi _{r_k}=\phi$.
 (Recall that $(m_k,\phi _{r_k})$ denotes an enumeration of the
pairs $(m,\phi ) \in \mathbb{N} \times \mathcal{D}$.) Then,
passing to the limit and using the assumed continuity of $s \to
u(s,\omega )$, the left-hand side of this equality converges
to $\langle \phi , u(s,\omega ) \rangle _V$ and so
the right-hand side, $\gamma _k(t,\omega )$, must also
converge and for each $\omega$. Since the right-hand side is a product
measurable function of $(t,\omega ), $ it follows that the
pointwise limit is also product measurable. Hence, $(t,\omega )
\to \langle \phi , u(t,\omega ) \rangle _V$
is product measurable for each $\phi \in \mathcal{D}$. Since $\mathcal{D}$
is a dense set, it follows that $(t,\omega ) \to
\langle \phi , u(t,\omega ) \rangle _V$ is $
\mathcal{P}$ measurable for all $\phi \in V$ and so by the Pettis theorem,
\cite{yos78}, $(t,\omega ) \to u(t,\omega )$
is $\mathcal{P}$ measurable into $V'$. This completes the proof.
\end{proof}

Actually, one can say more about the measurability of the approximating
sequence and in fact, we can obtain one for which $\omega \to
u_{n(\omega ) }(t,\omega )$ is also $\mathcal{F}$
measurable.

\begin{lemma}\label{lemma7}
 Suppose that $u_{n(\omega ) }\to u$
weakly in $L^{p'}([ 0,T] ;V')$,
where $u$ is product measurable, and $\{ u_{n(\omega )
}\} $ is a subsequence of $\{ u_n\} $, such that there
exists a set of measure zero $N \subseteq \Omega$ and
\[
\sup_{t\in [ 0,T] }\| u_n(t,\omega )
\| _{V'}<C(\omega ) ,\quad \text{for }\omega
\notin N.
\]
Then, there exists a subsequence of $\{ u_n\} $, denoted
as $\{ u_{k(\omega ) }\} $, such that $u_{k(\omega
) }\to u$ weakly in $L^{p'}([ 0,T]
;V')$, $\omega \to k(\omega )$ is
$\mathcal{F}$ measurable, and $\omega \to u_{k(\omega )
}(t,\omega )$ is also $\mathcal{F}$ measurable, for
each $\omega \notin N$.
\end{lemma}

We introduce the notation
\[
\mathcal{V}\equiv L^{p}([ 0,T ] ;V),\quad \mathcal{V}
'\equiv L^{p'}([ 0,T] ;V').
\]


\begin{proof}
Assume that $f,g\in \mathcal{V}'$ and let $\{ \phi
_k\} $ be a countable dense subset of $\mathcal{V}$. Then, a bounded
set in $\mathcal{V}'$ with the weak topology can be considered a
complete metric space using the metric
\[
d(f,g) \equiv \sum_{j=1}^{\infty }2^{-j}\frac{|
\langle \phi _k,f-g\rangle _{\mathcal{V}}| }{
1+| \langle \phi _k,f-g\rangle _{\mathcal{V}
}| }.
\]
Now, let $k(\omega )$ be the first index of $\{
u_n\} $ that is at least as large as $k$ and such that
\[
d(u_{k(\omega ) },u) \leq 2^{-k}.
\]
Such an index exists because there exists a convergent sequence $u_{n(
\omega ) }$ that converges weakly to $u$. In fact,
\[
\{ \omega :k(\omega ) =l\} =\{ \omega :d(
u_{l},u) \leq 2^{-k}\} \cap {\cap _{j=1}^{l-1}}\{ \omega
:d(u_j,u) >2^{-k}\}.
\]
Since $u$ is product measurable and each $u_{l}$ is also product measurable,
these are all measurable sets with respect to $\mathcal{F}$ and so $\omega
\to k(\omega )$ is $\mathcal{F}$ measurable. Now, we
have that $u_{k(\omega ) }\to u$ weakly in $L^{p'}([ 0,T] ;V')$, for each $\omega $, and
each function is $\mathcal{F}$ measurable because
\[
u_{k(\omega ) }(t,\omega ) =\sum_{j=1}^{\infty }
\mathcal{X}_{[ k(\omega ) =j] }u_j(t,\omega
),
\]
and every term in the sum is $\mathcal{F}$ measurable.
\end{proof}

Finally, when all the functions have values in a separable
Hilbert space $H$, the same arguments yield the following theorem
noting that the norms in \eqref{21a} and \eqref{24} are different.

\begin{theorem}\label{thm8}
Let $H$ be a real separable Hilbert space. Let the functions
$t\to u_n(t,\omega )$, for $n\in \mathbb{N}$, be
in $L^2([ 0,T] ;H)$ and $(t,\omega )
\to u_n(t,\omega )$ be $\mathcal{B}([ 0,T
] ) \times \mathcal{F}\equiv \mathcal{P}$ measurable into $H$.
Suppose there is a set of measure zero $N \subseteq \Omega$ such that
if $\omega \notin N$, then for all $n$,
\begin{equation}
\sup_{t\in [ 0,T] }| u_n(t,\omega )
| _{H}\leq C(\omega ). \label{24}
\end{equation}
Further, suppose that for each $\omega \notin N$, each subsequence
of $\{ u_n\} $ has a subsequence that converges weakly in
$L^2([ 0,T] ;H)$ to $u(\cdot ,\omega )
\in L^2([ 0,T] ;H)$ such that $t\to
u(t,\omega )$ is weakly continuous into $H$. Then, there exists
a product measurable function $u$ such that $t\to u(t,\omega
)$ is weakly continuous into $H$. Moreover, there exists, for each
$\omega \notin N$, a subsequence $u_{n(\omega ) }$ such
that $u_{n(\omega ) }(\cdot ,\omega ) \to u(
\cdot ,\omega )$ weakly in $L^2([ 0,T] ;H)$.
\end{theorem}

\section{Measurability for delay-differential equations}
\label{sec:finite}

In this section we use our main theorem to establish a Peano-type existence
theorem that provides a solution of the differential equation that retains
its product measurability. In particular, this result applies to general
second order ordinary differential equations with one delay. It is an
interesting example of the above theory and will be used in the Section 4 to
show the convergence of the Galerkin method. Moreover, although the material
on filtrations is not needed below, we include it because it is of interest
and will be used in the future. We note that a \textit{filtration } on $
[ 0,T] $ consists of a family of $\sigma $-algebras with $
\mathcal{F}_{t}$ for each $t\in [ 0,T] $ such that for $s<t,
\mathcal{F}_{s}\leq \mathcal{F}_{t}$. In applications to stochastic
integration, $\mathcal{F}_{t}$ is often chosen as $\sigma (W(
s) :s\leq t) $, the smallest $\sigma $-algebra for which each
one of the $W(s)$ is measurable, where $t\to W(t)$ is a Wiener process.

We recall that $\mathcal{P}\equiv \mathcal{B}([ 0,T]
) \times \mathcal{F}$, $\Omega$ is the sample space, and $M(\omega)$
and $C(\omega)$ represent constants that depend only on the problem data and
$\omega$.

Our first result deals with the case when $\mathbf{N}$ and $\mathbf{f}$ are
bounded functions. We assume that for fixed $\mathbf{u,v,w}$ in $\mathbb{R}
^{d}$,
\begin{equation}
(t,\omega ) \to \mathbf{N}(t,\mathbf{u},\mathbf{
v,w },\omega ) \label{measurable}
\end{equation}
is product measurable and that\ $(t,\mathbf{u,v,w}) \to
\mathbf{N}(t,\mathbf{u},\mathbf{v,w},\omega )$ is continuous.
We make this assumption so that if $\mathbf{u,v,w}$ are each product
measurable functions, then so is
\[
(t,\omega ) \to \mathbf{N}(t,\mathbf{u}(
t,\omega ) ,\mathbf{v}(t,\omega ) \mathbf{,w}(
t,\omega ) ,\omega ).
\]
This follows by using approximations with simple functions.

Our result is as follows.

\begin{theorem}\label{prop9}
Suppose that the function $\mathbf{N}:[0,T]\times \mathbb{R}
^{3d}\times \Omega \to \mathbb{R}^{d}$ is such that for
$\mathbf{u,v,w}\in \mathbb{R}^{d}$, $t\in [ 0,T] $ and $\omega \in \Omega $
the mapping $(t,\omega ) \to \mathbf{N}(t,\mathbf{u},\mathbf{v,w},\omega )$ is product measurable. Also, suppose that
the mapping $(t,\mathbf{u,v,w}) \to \mathbf{N}(t,
\mathbf{u},\mathbf{v,w},\omega )$ is continuous and that
\begin{equation}
|\mathbf{N}(t,\mathbf{u},\mathbf{v,w},\omega ) |\leq M(
\omega ) , \label{31}
\end{equation}
uniformly in $(t,\mathbf{u,v,w})$. Let $\mathbf{f}$ be $
\mathcal{P}$ measurable and $\ \mathbf{f}(\cdot ,\omega ) \in
L^2([ 0,T] ;\mathbb{R}^{d}) $.

Then, for $h\geq 0$, there exists a $\mathcal{P}$ measurable solution
$\mathbf{u}$ to the integral equation
\begin{equation}
\mathbf{u}(t,\omega ) +\int_0^{t}\mathbf{N}(s,\mathbf{u}
(s,\omega ),\mathbf{u}(s-h,\omega ) ,\mathbf{w}(s,\omega
) ,\omega ) ds=\mathbf{u}_0(\omega )+\int_0^{t}\mathbf{f}
(s,\omega ) ds. \label{32}
\end{equation}
Here, $\mathbf{u}_0\ $has values in $\mathbb{R}^{d}$ and is $\mathcal{F}$
measurable, $\mathbf{u}(s-h,\omega ) =\mathbf{u}_0(
\omega )$ if $s-h<0$ and for $\mathbf{w}_0$ a given $\mathcal{F}$
measurable function,
\[
\mathbf{w}(t,\omega ) \equiv \mathbf{w}_0(\omega )
+\int_0^{t}\mathbf{u}(s,\omega ) ds.
\]
\end{theorem}

\begin{proof}
The proof is based on the use of the delay operator $\tau _{\delta}$ defined
as follows. For $\delta >0$, we let
\[
\tau _{\delta }\mathbf{u}(s) \equiv
\begin{cases}
\mathbf{u}(s-\delta ) & \text{if }s>\delta, \\
\mathbf{0} & \text{if }s-\delta \leq 0.
\end{cases}
\]
 Now, let $\mathbf{u}_n$ be the solution of the equation
\begin{align*}
&\mathbf{u}_n(t,\omega ) +\int_0^{t} \mathbf{N}(s,\tau
_{1/n}\mathbf{u}_n(s,\omega ),\mathbf{u}_n(s-h,\omega )
,\tau _{1/n}\mathbf{w}_n(s,\omega ),\omega ) ds\\
&=\mathbf{u}_0(\omega )+\int_0^{t}\mathbf{f}(s,\omega ) ds.
\end{align*}
It follows that $(t,\omega ) \to \mathbf{u}_n(
t,\omega )$ is $\mathcal{P}$ measurable. The assumptions on
$\mathbf{N }$ guarantee that for a fixed $\omega$ the family of functions
$\{ \mathbf{u}_n(\cdot,\omega ) \} $ is uniformly
bounded, indeed,
\[
\sup_{t\in \lbrack 0,T]}| \mathbf{u}_n(t,\omega )
| \leq | \mathbf{u}_0(\omega )|
+\int_0^{T}M(\omega ) ds+\int_0^{T}| \mathbf{f}
(s,\omega ) | ds \equiv C(\omega ).
\]
It is also equicontinuous since for $s<t$,
\begin{align*}
| \mathbf{u}_n(t,\omega ) -\mathbf{u}_n(
s,\omega ) |
&\leq \int_{s}^{t}| \mathbf{N}(
r,\tau _{1/n}\mathbf{u}_n(r,\omega ),\mathbf{u}_n(r-h,\omega
) ,\tau _{1/n}\mathbf{w}_n(r,\omega ) ,\omega )
| dr \\
&\quad +\int_{s}^{t}| \mathbf{f}(r,\omega ) | dr\\
&\leq C(\omega ,\mathbf{f}) | t-s| ^{1/2}.
\end{align*}
Therefore, by the Ascoli-Arzela theorem, for each $\omega$ there exist a
subsequence $\tilde{n}(\omega )$, which depends on $\omega $, and a function
$\mathbf{\tilde{u}}(t,\omega )$ such that
\[
\mathbf{u}_{\tilde{n}(\omega )}(t,\omega ) \to \mathbf{
\tilde{u}}(t,\omega ) \text{ uniformly in }C([ 0,T
] ;\mathbb{R}^{d}) .
\]
This verifies that the assumptions of Theorem \ref{thm8} hold. It follows
that there exists a function $\mathbf{\bar{u}}$ that is product measurable
and a subsequence $\{ \mathbf{u}_{n(\omega ) }\} $,
for each $\omega $, such that
\[
\lim_{n(\omega )\to \infty }\mathbf{u}_{n(\omega )
}(\cdot ,\omega ) =\mathbf{\bar{u}}(\cdot ,\omega )
\text{ weakly in }L^2([ 0,T] ;\mathbb{R}^{d})
\]
and that $t\to \mathbf{\bar{u}}(t,\omega )$ is
continuous, since weak continuity is the same as continuity in $\mathbb{R}
^{d}$. The same argument given above applied to the $\mathbf{u}_{n(
\omega ) }$, for a fixed $\omega $, yields a further subsequence,
denoted as $\{ \mathbf{u}_{\bar{n}(\omega ) }(\cdot
,\omega ) \} $ which converges uniformly to a function
 $\mathbf{u }(\cdot ,\omega )$ on $[ 0,T] $. So
$\mathbf{\bar{u }} (t,\omega ) =\mathbf{u}(t,\omega )$ in
$L^2([ 0,T] ;\mathbb{R}^{d})$. Since both of these
functions are continuous in $t$, they must be equal for all $t$. Hence,
$(t,\omega ) \to \mathbf{u}(t,\omega )$ is product
measurable. Passing to the limit in the equation solved by
$\{\mathbf{u }_{\bar{n} (\omega ) }(\cdot ,\omega )\} $ and using
 the dominated convergence theorem, we obtain
\[
\mathbf{u}(t,\omega ) -\mathbf{u}_0(\omega )
+\int_0^{t}\mathbf{N}(s,\mathbf{u}(s,\omega ),\mathbf{u}(
s-h,\omega ) ,\mathbf{w}(s,\omega ) ,\omega )
ds=\int_0^{t}\mathbf{f}(s,\omega ) ds.
\]
Thus $t\to \mathbf{u}(t,\omega )$ is a product
measurable solution of the integral equation.
\end{proof}

The theorem provides the existence of the approximate solutions needed in
the next theorem in which the assumption that the integrand is bounded is
replaced with an appropriate estimate. However, first we mention the
following elementary lower-bound inequality that is used below.

\begin{lemma}\label{lemma10}
 Assume that $\mathbf{w}(t)=\mathbf{w}_0(\omega
) +\int_0^{t}\mathbf{u}(s,\omega ) ds$, define $\mathbf{v }$ as
\[
\mathbf{v}(t) =
\begin{cases}
\mathbf{u}(t-h) & \text{if }t\geq h, \\
\mathbf{u}_0 & \text{otherwise},
\end{cases}
\]
and that the following estimate holds true,
\[
(\mathbf{N}(t,\mathbf{u},\mathbf{v,w},\omega ) ,\mathbf{u}
) \geq -C(t,\omega ) -\mu (| \mathbf{u}
| ^2+| \mathbf{v}| ^2+| \mathbf{w}
| ^2).
\]
Then,
\[
\int_0^{t}(\mathbf{N}(t,\mathbf{u},\mathbf{v,w},\omega )
,\mathbf{u}) ds
\geq -C\Big(C(\omega )
+\int_0^{t}| \mathbf{u}| ^2ds\Big),
\]
for some constant $C$ depending on the initial data but not on $\mathbf{u}$.
\end{lemma}

\begin{proof}
We have
\begin{align*}
\int_0^{t}| \mathbf{u}(s-h) |^2ds
&=\int_0^{h}| \mathbf{u}_0|
^2ds+\int_{h}^{t}| \mathbf{u}(s-h) | ^2ds \\
&=| \mathbf{u}_0| ^2h+\int_0^{t-h}| \mathbf{u}(s) | ^2ds\\
&\leq | \mathbf{u} _0| ^2h+\int_0^{t}| \mathbf{u}(s)| ^2ds,
\end{align*}
when $t\geq h$ and when $s<h$, the integral is dominated by
\[
| \mathbf{u}_0| ^2t\leq | \mathbf{u}
_0| ^2h\leq | \mathbf{u}_0|
^2h+\int_0^{t}| \mathbf{u}(s) | ^2ds.
\]
Next, using the usual inequalities yields
\begin{align*}
\int_0^{t}| \mathbf{w}(s) | ^2ds
&\leq \int_0^{t}\big| \mathbf{w}_0+\int_0^{s}\mathbf{u}(r)
dr\big| ^2ds \\
&\leq \int_0^{t}\Big(| \mathbf{w}_0|
^2+2| \mathbf{w}_0| \big| \int_0^{s}\mathbf{u}
(r) dr\big| +\big| \int_0^{s}\mathbf{u}(
r) dr\big| ^2\Big) ds \\
&\leq T| \mathbf{w}_0| ^2+T| \mathbf{w}
_0| ^2+\int_0^{t}\big| \int_0^{s}\mathbf{u}(
r) dr\big| ^2ds+\int_0^{t}\big| \int_0^{s}\mathbf{u}
(r) dr\big| ^2ds \\
&\leq 2T| \mathbf{w}_0| ^2+2\int_0^{t}\Big(
\int_0^{s}| \mathbf{u}(r) | dr\Big) ^2ds\\
&\leq 2T| \mathbf{w}_0|
^2+2\int_0^{t}s\int_0^{s}| \mathbf{u}(r)
| ^2drds \\
&\leq 2T| \mathbf{w}_0|^2+2T^2\int_0^{t}| \mathbf{u}(r) |^2dr.
\end{align*}
These estimates lead directly to the claimed result.
\end{proof}

We now state a more general result in which $\mathbf{N}$ is only bounded
from below, which is the main result in this section.

\begin{theorem}\label{thm11}
Suppose that the function $\mathbf{N}:[0,T]\times \mathbb{R}^{3d}\times
\Omega \to \mathbb{R}^{d}$ is such that for
$\mathbf{u,v,w}\in \mathbb{R}^{d},t\in [ 0,T] $ and $\omega \in \Omega $
the mapping $(t,\omega ) \to \mathbf{N}(t,\mathbf{u },\mathbf{v,w},\omega )$
is product measurable. Also, suppose
\[
(t,\mathbf{u,v,w}) \to \mathbf{N}(t,\mathbf{u},
\mathbf{v,w},\omega )
\]
is continuous, and there are a nonnegative function
$C(\cdot ,\omega) \in L^{1}([ 0,T] )$ and a positive constant $
\mu $ such that
\begin{equation}
(\mathbf{N}(t,\mathbf{u},\mathbf{v,w},\omega ) ,\mathbf{u}
) \geq -C(t,\omega ) -\mu (| \mathbf{u}
| ^2+| \mathbf{v}| ^2+| \mathbf{w}
| ^2) . \label{33}
\end{equation}
Moreover, let $\mathbf{f}$ be product measurable and $\mathbf{f}(\cdot
,\omega ) \in L^2([ 0,T] ;\mathbb{R}^{d})$.

Then, for each $h\geq 0$, there exists a product measurable solution
$\mathbf{u}$ to the integral equation
\begin{equation}
\mathbf{u}(t,\omega ) +\int_0^{t} \mathbf{N}(s,\mathbf{u}
(s,\omega ),\mathbf{u}(s-h,\omega ) ,\mathbf{w}(s,\omega
) ,\omega ) ds=\mathbf{u}_0(\omega )+\int_0^{t}\mathbf{f}
(s,\omega ) ds, \label{34}
\end{equation}
where $\mathbf{u}_0\ $has values in $\mathbb{R}^{d}$ and is $\mathcal{F}$
measurable. Here, $\mathbf{u}(s-h,\omega ) \equiv \mathbf{u}
_0(\omega )$ for all $s-h\leq 0$ and for $\mathbf{w}_0$ a
given $\mathcal{F}$ measurable function,
\[
\mathbf{w}(t,\omega ) \equiv \mathbf{w}_0(\omega )
+\int_0^{t}\mathbf{u}(s,\omega ) ds
\]
\end{theorem}

\begin{proof}
The idea of the proof is to bound the variables, so that $\mathbf{N}$ is
bounded, use Theorem \ref{prop9} to obtain product measure solutions, and
pass to the limit when the variables are allowed be be unbounded.

Let $P_m$ denote the projection onto the closed ball
$\overline{B( \mathbf{0},9^{m}) }\subset\mathbb{R}^d$. Then, it follows from Theorem
\ref{prop9} that there exists a product measurable solution $\mathbf{u}_m$
of the integral equation
\begin{align*}
&\mathbf{u}_m(t,\omega ) +\int_0^{t}\mathbf{N}(s,P_m
\mathbf{u}_m(s,\omega ),P_m\mathbf{u }_m(s-h,\omega ),P_m\mathbf{w}
_m(s,\omega ) ,\omega ) ds\\
&=\mathbf{u}_0(\omega)+s\int_0^{t}\mathbf{f}(s,\omega ) ds.
\end{align*}
Next, we define a stopping time
\[
\tau _m(\omega ) \equiv \inf \big\{ t\in [ 0,T] :|
\mathbf{u }_m(t,\omega ) | ^2+| \mathbf{w}
_m(t,\omega ) | ^2>2^{m}\big\},
\]
where we use the convention that $\inf\, \{\emptyset\} = T$. Localizing with
the stopping time,
\begin{align*}
&\mathbf{u}_m^{\tau _m}(t,\omega ) +\int_0^{t}\mathcal{X}_{
[ 0,\tau _m] }\mathbf{N }(s,\mathbf{u}_m^{\tau
_m}(s,\omega ),\mathbf{u}_m^{\tau _m}(s-h,\omega ),\mathbf{w}
_m^{\tau _m}(s,\omega ) ,\omega ) ds\\
&=\mathbf{u}
_0(\omega ) +\int_0^{t}\mathcal{X}_{[ 0,\tau _m]
}\mathbf{f} (s,\omega ) ds.
\end{align*}
Note that the stopping time allowed to eliminate the projection operator in
the equation. Then, we obtain
\begin{align*}
&\frac{1}{2}| \mathbf{u}_m^{\tau _m}(t,\omega )| ^2\\
&+\int_0^{t}\Big(\mathcal{X}_{[ 0,\tau _m] }
\mathbf{N} \big(s,\mathbf{u}_m^{\tau _m}(s,\omega ),\mathbf{u}
_m^{\tau _m}(s-h,\omega ),\mathbf{w}_m^{\tau _m}(s,\omega
) ,\omega \big) ,\mathbf{u}_m^{\tau _m}(s,\omega )\Big) ds \\
&=\frac{1}{2}| \mathbf{u}_0(\omega )|
^2+\int_0^{t}\mathcal{X}_{[ 0,\tau _m] }(\mathbf{f}
(s,\omega ) ,\mathbf{u}_m^{\tau _m}(s,\omega )) ds.
\end{align*}
Therefore,
\begin{align*}
\frac{1}{2}| \mathbf{u}_m^{\tau _m}(t,\omega )| ^2
&\leq \int_0^{t} \Big(\mu \Big(| \mathbf{u}
_m^{\tau _m}(s,\omega )| ^2+| \mathbf{u}_m^{\tau
_m}(s-h,\omega )| ^2+| \mathbf{w}_m^{\tau
_m}(s,\omega ) | ^2\Big) \\
&\quad +C(s,\omega ) +\frac{1}{2}| \mathbf{f}(s,\omega) | ^2\Big) ds
+\frac{1}{2}\int_0^{t}\vert \mathbf{u }_m^{\tau _m}(s,\omega )| ^2ds
 + \frac{1}{2}|\mathbf{u}_0(\omega )| ^2.
\end{align*}
We note that
\[
| \mathbf{u}_0| ^2h+\int_0^{t}| \mathbf{u}_n^{\tau _n}(s) | ^2ds
\geq \int_0^{t}| \mathbf{u}_n^{\tau _n}(s-h,\omega )|^2ds,
\]
and
\begin{align*}
\int_0^{t}| \mathbf{w}_n^{\tau _n}(s,\omega )| ^2ds
&=\int_0^{t}\big| \mathbf{w}_0+\int_0^{s}
\mathcal{X}_{[ 0,\tau _n] }\mathbf{u}_n(r)
dr\big| ^2ds \\
&=\int_0^{t}\big| \mathbf{w}_0+\int_0^{s}\mathcal{X}_{[
0,\tau _n] }\mathbf{u}_n^{\tau _n}(r) dr\big|^2ds \\
&\leq C(\mathbf{w}_0(\omega ) )
+CT\int_0^{t}| \mathbf{u}_n^{\tau _n}| ^2ds.
\end{align*}
Using now the Gronwall inequality yields
\begin{align*}
| \mathbf{u}_m^{\tau _m}(t,\omega ) |^2
&\leq C(\mathbf{u}_0(\omega ),\mathbf{w}_0(\omega
) ,\mu ,\| C(\cdot ,\omega ) \|
_{L^{1}([ 0,T] ;\mathbb{R}^{d}) },T,\|
\mathbf{f}(\cdot ,\omega ) \| _{L^2([ 0,T
] ;\mathbb{R}^{d}) }) \\
&= C(\omega ).
\end{align*}
Thus, it follows from the definition of the stoping time that for a.e.
$\omega ,\tau _m=T$ for all $m$ large enough, say for $m\geq
M(\omega )$ where $C(\omega ) \leq
2^{M(\omega ) }$. Next, we define the functions
\[
\mathbf{y}_n(t,\omega ) \equiv \mathbf{u}_n^{\tau
_n}(t,\omega ),
\]
which are product measurable and satisfy
\begin{align*}
&\mathbf{y}_n(t,\omega ) +\int_0^{t}\mathcal{X}_{[
0,\tau _n] }\mathbf{N}\Big(s, \mathbf{y}_n(s,\omega ),\mathbf{y}
_n(s-h,\omega ) ,\mathbf{w} _0(\omega )
+\int_0^{s}\mathbf{y}_n(r,\omega ) dr,\omega \Big) ds \\
&=\mathbf{u}_0(\omega )+ \int_0^{t}\mathcal{X}_{
[ 0,\tau _n] }\mathbf{f}(s,\omega ) ds.
\end{align*}
So each function is also continuous in $t$. Since $\tau _n=T$ for large
enough $n$, it follows that
\begin{align*}
&\mathbf{y}_n(t,\omega ) +\int_0^{t}\mathbf{N}\Big(s,
\mathbf{y}_n(s,\omega ),\mathbf{y} _n(s-h,\omega ) ,\mathbf{
w }_0(\omega ) +\int_0^{s}\mathbf{y}_n(r,\omega
) dr,\omega \Big) ds\\
&= \mathbf{u}_0(\omega)+\int_0^{t}\mathbf{f}(s,\omega ) ds.
\end{align*}
Also, these functions satisfy the inequality
\begin{equation}
\sup_{t\in \lbrack 0,T]}| \mathbf{y}_n(t,\omega )
| ^2\leq C(\omega ) \leq 2^{M(\omega )}<9^{M(\omega )}, \label{35}
\end{equation}
where the constants on the right-hand side do not depend on $n$. Thus, for
fixed $\omega $, we can regard $\mathbf{N}$ as bounded and the same
reasoning used in Theorem \ref{prop9} involving the Ascoli-Arzela theorem
implies that every subsequence has a further subsequence that converges to a
solution of the integral equation for that $\omega $. Hence, it is
continuous into $\mathbb{R} ^{d}$. It follows from the measurable selection
theorem, Theorem \ref{thm1}, that there exists a product measurable function
$\mathbf{u}$ that is continuous in $t$ such that
$\mathbf{u}(\cdot,\omega ) =\lim_{n(\omega )\to \infty }
\mathbf{y}_{n(\omega ) }(\cdot ,\omega )$ in
$L^2([ 0,T ] ;\mathbb{R}^{d})$. By the reasoning above, there is a further
subsequence, denoted the same way, for which $\lim_{n\to \infty }
\mathbf{y}_{n(\omega ) }$ in $C([ 0,T] ;\mathbb{
\ R}^{d})$ solves the integral equation for a fixed $\omega $.
Thus $\mathbf{u}$ is a product measurable solution to the integral
equation \eqref{34} as claimed.
\end{proof}

We made use of estimate \eqref{33} in the proof of this theorem. However,
all that is really needed is the following simpler condition.

\begin{corollary}\label{corollary12}
Suppose that the function $\mathbf{N}:[0,T]\times
\mathbb{R}^{3d}\times \Omega \to \mathbb{R}^{d}$ is such that
for $\mathbf{u,v,w}\in \mathbb{R}^{d},t\in [ 0,T] $ and
$\omega \in\Omega $ the mapping $(t,\omega ) \to \mathbf{N}(
t, \mathbf{u},\mathbf{v,w},\omega )$ is product measurable. Also,
suppose that $(t,\mathbf{u,v,w}) \to \mathbf{N}(t,
\mathbf{u},\mathbf{v,w},\omega )$ is continuous. Moreover, for
each $\omega $, for each solution $\mathbf{u}(\cdot ,\omega )$
of the integral equation
\begin{equation}
\mathbf{u}(t,\omega ) +\int_0^{t}\mathbf{N}(s,\mathbf{u}
(s,\omega ),\mathbf{u}(s-h,\omega ) ,\mathbf{w}(s,\omega
) ,\omega ) ds=\mathbf{u}_0(\omega )+\int_0^{t}\mathbf{f}
(s,\omega ) ds, \label{36}
\end{equation}
there exists an estimate of the form
\begin{equation}
\sup_{t\in [ 0,T] }| \mathbf{u}(t,\omega )
| \leq C(\omega ) <\infty . \label{37}
\end{equation}
Moreover, let $\mathbf{f}$ be product measurable and $\mathbf{f}(\cdot
,\omega ) \in L^{1}([ 0,T] ;\mathbb{R}^{d})$; $
\mathbf{u}_0$ has values in $\mathbb{R}^{d}$ and is $\mathcal{F}$
measurable and $\mathbf{u}(s-h,\omega ) \equiv \mathbf{u}
_0(\omega )$ whenever $s-h\leq 0$; and
\[
\mathbf{w}(t,\omega ) \equiv \mathbf{w}_0(\omega )
+\int_0^{t}\mathbf{u}(s,\omega ) ds,
\]
where $\mathbf{w}_0$ is a given $\mathcal{F}$ measurable function.

Then, for $h\geq 0$, there exists a product measurable solution $\mathbf{u}$
of the integral equation \eqref{34}.
\end{corollary}

We note that the same conclusions apply when there is no dependence of the
integrand on $\mathbf{u}(s-h,\omega )$, that is, there are no
delays, or on $\mathbf{w }(s,\omega )$.

\section{A contact problem with friction}
\label{sec:contact}

We apply our theoretical result, Theorem\,1, to an important problem of
dynamic contact with friction between a viscoelastic body and a deformable
foundation in which the coefficient of friction depends on the relative slip
speed. The problem without randomness was studied in \cite{KS02}, and the
novelty here is that the gap between the body and the foundation is assumed
to be a random variable and so is the foundation's velocity. These two
changes make the model much more realistic since in engineering applications
both can be determined only up to relatively large tolerances.

The setting of the problem is depicted in Figure \ref{fig1}. A viscoelastic
body occupies the domain $U\subseteq \mathbb{R}^{d}$ (where $d=2,3$ in
applications) that is a bounded open subset with Lipschitz boundary $\Gamma
=\partial U$. The boundary $\Gamma$ consists of three parts: $\Gamma _{D}$
where a Dirichlet data is prescribed, $\Gamma _{N}$ where a Neumann
condition holds, and the potential contact surface with the foundation
$\Gamma _C$. We denote by $\mathbf{n}$ the outer unit
normal to $U$ on $\Gamma$. Moreover, when the foundation is planar, we assume that it
moves with velocity $\mathbf{v}^{*}$. We also let $d=3$ as
the 2D case is somewhat simpler. Finally, in this section $\mu$ denotes the
friction coefficient, and not the Lebesgue measure.

\begin{figure}[ht]
\begin{center}
\begin{picture}(222,140)(108,-6)
\thinlines \put(137,115){$\Gamma_D$}
 \put(290,70){$\Gamma_{N}$}
\thinlines \put(158,24){$\Gamma_C$}
 \put(213,17){\vector(0,-1){21}}
 \put(216,-1){${\mathbf{n}}$}
 \put(82,-6){$Foundation$}
 \put(175,-3){\vector(1,0){24}}
 \put(302,12){$g\,-gap$ }
 \put(302,35){\line(2,-1){12}}
 \put(280,125){\line(0,1){10}}
 \put(86,38){\line(1,0){10}}
 \put(160,70){$U \;- Body$}
 \put(162,-4){$\mathbf{v}^{*}$}
 \put(80,8){\line(1,0){280}}
 \put(201,74){\oval(220,114)}
\end{picture}
\end{center}
\caption{$\Gamma_C$ is the contact surface and $g$ is the gap}
\label{fig1}
\end{figure}

We denote by $\mathbf{u}=\mathbf{u}(\mathbf{x}, t)$ the displacement vector
for $\mathbf{x}\in \overline{U}$ and $t\in [0, T]$, by
 $\boldsymbol{\varepsilon} =(\varepsilon_{ij})$ the linearized strain tensor, and
by $\boldsymbol{\sigma}=(\sigma_{ij})$ the stress tensor; here
and below $i,j,k,l=1,2,3$. A dot above a symbol denotes the partial time derivative,
while an index following a coma indicates partial derivative with respect to
the indicated spatial variable, i.e. $u_{i,j}=\partial u_{i}/\partial x_j$.
 Moreover, summation over an index that appears twice is implied.

We assume that the material is linearly viscoelastic with short-term memory,
with constitutive relation
\[
\boldsymbol{\sigma}= A\boldsymbol{\varepsilon}(\mathbf{u}) +C
\boldsymbol{\varepsilon}(\dot{\mathbf{u}}),
\]
where $A=(a_{ijkl})$ is the elasticity tensor and $C=(c_{ijkl})$ the
viscosity tensor, both described in more detail below. The body is being
acted upon by the force density $\rho \mathbf{f}$, and for the sake of
simplicity we rescale the variables so that the material density is $\rho=1$.
 On $\Gamma_{D}$ the body is clamped so that $\mathbf{u}=0$, and a
prescribed traction $\mathbf{t}$ acts on $\Gamma_{N}$.

The dynamic equations of motion and the initial and boundary conditions are
as follows.
\begin{gather}
\ddot{\mathbf{u}}=\operatorname{Div}\boldsymbol{\sigma}(\mathbf{u},
\mathbf{\dot{u}})+\mathbf{f},\quad (t,\mathbf{x})\in (0,T)\times U,
\label{a1.1}
\\
\mathbf{u}(0,\mathbf{x})=\mathbf{u}_0(\mathbf{x}),\quad \mathbf{\dot{u}}
(0,\mathbf{x})=\mathbf{v}_0(\mathbf{x}),\quad \mathbf{x}\in U,
\label{a1.2}
\\
\mathbf{u}(t,\mathbf{x})=0,\quad (t,\mathbf{x})\in (0,T)\times \Gamma
_{D};\quad \boldsymbol{\sigma}(t,\mathbf{x})\cdot
\mathbf{n}=\mathbf{t},\quad (t,\mathbf{x})\in (0,T)\times
\Gamma _{N}. \label{a1.3}
\end{gather}
Here, $\mathbf{n}$ is the outer unit normal to $U$ on $\Gamma $.
\vskip4pt Next, we describe the contact conditions on $\Gamma _C$. To that
end we need the normal and tangential components and parts of the vectors on
the surface, so we let
\begin{gather*}
u_n=\mathbf{u}\cdot \mathbf{n},\quad
\mathbf{u}_{\tau }=\mathbf{u}-(\mathbf{u}\cdot \mathbf{n})\mathbf{n},
\\
\sigma _n=\sigma _{ij}n_jn_{i},\quad
\sigma _{\tau i}=\sigma_{ij}n_j-\sigma _nn_{i},
\end{gather*}
written more simply, $\sigma _n=\mathbf{n}\cdot
\boldsymbol{\sigma}\cdot \mathbf{n}$ and $\mathbf{\sigma }
_{\tau }=\boldsymbol{\sigma}\mathbf{n}-\sigma _n
\mathbf{n}$.

We assume that contact is described by the normal compliance condition (see,
e.g., \cite{Mar87,KMS88,KMS89, kut97,KS04-1,RSS98-1, SST04book} and the
references therein) and the friction process by an appropriately modified
condition of the Coulomb-type, thus, for $\mathbf{x}$ $\in $ $\Gamma _C$,
we assume
\begin{equation}
\sigma _n=-p(u_n-g), \label{a1.4}
\end{equation}
where $p(\cdot )$ is the normal compliance function, assumed to be
nonnegative and to vanish when there is no contact, i.e., when the normal
displacement is less than the gap, $u_n\leq g$. The friction process is in
the tangential direction and there is relative motion only when the
tangential traction reaches the threshold of the \textit{friction bound}
denoted by $F\mu $, where $\mu $ is the friction coefficient, described
below. We refer the reader to \cite{KS02,SST04book} for further details. The
friction condition is
\begin{gather}
|\mathbf{\sigma }_{\tau }|\leq F(u_n-g)\mu (| \mathbf{
\dot{u}}_{\tau }-\mathbf{v}^{\ast }| ) ,\label{a1.5}
\\
|\mathbf{\sigma }_{\tau }|=\mu (| \mathbf{\dot{u}}_{\tau }-
\mathbf{v}^{\ast }| ) F(u_n-g)\quad \text{implies}\quad \mathbf{\dot{u}}_{\tau }-\mathbf{v}^{\ast }=-\lambda
\mathbf{\sigma }_{\tau }\mathbf{.} \label{a1.6}
\end{gather}
where $\lambda \geq 0$. Here, $\mathbf{v}^{\ast }$ is the
velocity of the foundation, which is known, and the friction coefficient
$\mu $ depends on the relative slip-rate, and is assumed to be a bounded
positive function having a bounded continuous derivative. It is
reasonable to assume that $\mu $ depends on $\mathbf{x}\in \Gamma _C$,
related to the pointwise roughness of the contact surface, however, we do
not make this dependence explicit for the sake of simpler notation.

The function $g$ represents the gap between the contact surface $\Gamma _C$
and a foundation along the direction $\mathbf{n}$. One of the
novel ingredients in this paper is that we allow the gap to be random, which
better describes real contact processes. Moreover, part of the novelty is
that we do not need to make any assumption on the sample space. It is often
the case that it is assumed to be the unit interval or the real line but no
such assumption is needed here. Indeed, the exact form of the sample space
does not enter the arguments. Therefore, in each application, one may
specify the appropriate sample space $\Omega$ freely. Therefore, we do not
specify $\Omega$ below. Thus,
\[
g=g(t,\mathbf{x}, \omega ),
\]
where $\omega \in \Omega$ and we assume that $(t,\mathbf{x},\omega
) \to g(\mathbf{x}, \omega )$ is $\mathcal{B}
([ 0,T] \times \Gamma _C) \times \mathcal{F}$
measurable, where $\mathcal{B}([ 0,T] \times \Gamma
_C)$ denotes the Borel sets of $[ 0,T] \times \Gamma
_C $. We assume that the gap is nonnegative (we do not consider
`shrink-fit' cases) and bounded, so
\[
0\leq g(t,\mathbf{x}, \omega ) \leq l_{*}<\infty,
\]
for all $(t,\mathbf{x},\omega )$ and some $l_{*}$. Additional
novelty in this work is that the motion of the foundation $
\mathbf{v}^*$ is assumed to be a stochastic process
\[
\mathbf{v}^*=\mathbf{v}^*(t,\mathbf{x}, \omega
),
\]
and is $\mathcal{B}([ 0,T] \times \Gamma _C) \times
\mathcal{F}$ measurable. We also assume that
$\mathbf{v}^{*}(t,\mathbf{\ x}, \omega )$ is uniformly bounded, and to
simplify the notation, we suppress the dependence on
$t,\mathbf{x} $ and $\omega $.

The normal compliance contact condition \eqref{a1.4} says that $\sigma _n$
the normal component of the traction density on $\Gamma _C$ is dependent
on the normal interpenetration of the body's surface asperities into those
of the foundation surface. Conditions \eqref{a1.5} and \eqref{a1.6} model
friction. They say that no sliding takes place until
$|\mathbf{\sigma } _{\tau }|$ reaches the friction bound $F(u_n-g)\mu (0)$ and
when this occurs, the tangential force density has a direction opposite to
the relative tangential velocity \eqref{a1.6}. The dependence of the
friction coefficient on the magnitude of the slip velocity, $|
\mathbf{\dot{u}}_{\tau }-\mathbf{v}^{\ast }| $ is
important and well documented (see, e.g., \cite{SST04book} and the
references therein) and so it has been included.

The two new features in this model are that the gap and the foundation's
velocity are random variables for each $\mathbf{x} \in \Gamma _C$. Our aim
is to show the measurability of the solutions. Thus, for a fixed $\omega$,
we have a friction problem that has been studied in the literature, and it
is the measurability which is of interest here. \vskip4pt

We assume the following on the functions $p$ and $F$. Both $p$
and $F$ are increasing and
\begin{gather}
\delta ^2r-K\leq p(r)\leq K(1+r),\quad r\geq 0,\; p(r)=0,\; r\leq 0,
\label{a1.8} \\
F(r)\leq K(1+r),\; r\geq 0,\quad F(r)=0,\; r\leq 0, \label{a1.9} \\
| \mu (r_1) -\mu (r_2) | \leq
Lip(\mu ) | r_1-r_2| ,\quad \| \mu \| _{\infty }\leq C, \label{a1.9a}
\end{gather}
and for $\psi =F,p$, and $r_1,r_2\geq 0$,
\begin{equation}
|\psi (r_1)-\psi (r_2)|\leq K|r_1-r_2|. \label{a1.10}
\end{equation}
One could consider more general growth conditions than these
(see \cite{kut97}), but we keep this part simple to emphasize the new stochastic
features.

The stress tensor is given by
\begin{equation}
\sigma _{ij}=A_{ijkl}u_{k_{,}l}+C_{ijkl}\dot{u}_{k,l}, \label{a1.11}
\end{equation}
where $A$ and $C$ are in $L^{\infty }(U)$ and for $B$ $=$ $A$ or $C$, we
have the following symmetries.
\begin{equation}
B_{ijkl}=B_{ijlk}\,,\quad B_{jikl}=B_{ijkl}\,,\quad B_{ijkl}=B_{klij}\,,
\label{a1.12}
\end{equation}
and we also assume that
\begin{equation}
B_{ijkl}H_{ij}H_{kl}\geq \varepsilon H_{rs}H_{rs} \label{a1.13}
\end{equation}
for all symmetric $H_{ij}$.

In the rest of this section, $V$ is a closed subspace of ($H^{1}(U))^3$
containing the space of test functions $(C_0^{\infty }(U))^3$;
$\rightharpoonup $ denotes weak convergence in the case of a reflexive
Banach space and weak$\ast $ convergence for a few examples of dual
spaces that are not reflexive, while $\to $ means strong convergence;
$\gamma $ denotes the trace map from $W^{1,2}(U)$ into $L^2(\Gamma )$;
$H$ denotes ($L^2(U))^3$ and we always identify $H$ and $H'$
to write
\[
V\subseteq H=H'\subseteq V',
\]
so that $(V,H,V')$ is a Gelfand triple. The duality pairing of $V$
and $V'$ is denoted by $\langle \cdot ,\cdot \rangle _V$.
We also define
\[
\mathcal{V}=L^2(0,T;V) ,\quad \mathcal{H}=L^2(
0,T,H) ,\quad \mathcal{V}'=L^2(0,T;V').
\]

We refer to \cite{ada75,Kuf77} for standard notation and properties of
Sobolev Spaces.

\subsection{The Abstract Problem}
\label{sec:abstract}

In this subsection we derive an abstract formulation of the problem that
allows us to use various tools and results from the theory of evolution
equations. However, first, we recall two theorems about compact sets of
functions found in Lions \cite{Lio69} and Simon \cite{Sim87}, respectively,
that we need below. These theorems apply for a fixed $\omega\in \Omega$.

\begin{theorem}\label{t2.3a}
Assume that the sets $W, U$ and $Y$ are such that $W\subseteq U\subseteq Y$,
and the inclusion map of $W$ into $U$ is compact
and the inclusion map of $U$ into $Y$ is continuous. Let $p\geq 1$, $q>1$,
and define
\[
S=\{\mathbf{u}\in L^{p}(0,T;W):\mathbf{u}'\in L^{q}(0,T;Y)
\text{ and } \|\mathbf{u}\|_{L^{p}(0,T;W)}+\|\mathbf{u}'\|_{L^{q}(0,T;Y)}<R\}.
\]
Then, $S$ is pre-compact in $L^{p}(0,T;U)$.
\end{theorem}

\begin{theorem}\label{t2.4}
Let $W,U$ and $Y$, and $p,q$, be as in Theorem \ref{t2.3a} and
let
\[
S=\{\mathbf{u}:\|\mathbf{u}(t)\|_{W}+\|\mathbf{u}'\|_{L^{q}(0,T;Y)}\leq R\quad
\text{for } t\in \lbrack 0,T]\}.
\]
Then, $S$ is pre-compact in $C(0,T;U)$.
\end{theorem}

Now, we obtain an abstract formulation of the problem 
\eqref{a1.1}--\eqref{a1.6}. We begin by defining the operators $M,A:V\to V'$
by
\begin{gather}
\langle M\mathbf{u},\mathbf{v}\rangle _V=\int_{U}C_{ijkl}\mathbf{u}_{k,l}
\mathbf{v}_{i,j}dx, \label{a2.15} \\
\langle A\mathbf{u},\mathbf{v}\rangle _V=\int_{U}A_{ijkl}\mathbf{u}_{k,l}
\mathbf{v}_{i,j}dx. \label{a2.16}
\end{gather}
Also, let the operator $\mathbf{v}\to P(\mathbf{u})$, mapping $
\mathcal{V}$ into $\mathcal{V}'$, be given by
\begin{equation}
\langle P(\mathbf{u}),\mathbf{w}\rangle _V=\int_0^{T}\int_{\Gamma
_C}p(u_n-g)w_n\,dS dt, \label{a2.17}
\end{equation}
where $dS$ is surface measure on $\Gamma $ and
\begin{equation}
\mathbf{u}(t)=\mathbf{u}_0+\int_0^{t}\mathbf{v}(s)ds, \label{a2.18}
\end{equation}
for $\mathbf{u}_0\in V$. We note that $P$ depends on $\mathbf{u}_0$ but
we suppress this in favor of simpler notation. Let
\[
\gamma _{\tau }^{\ast }:L^2\big(0,T;L^2(\Gamma _C)
^3\big) \to \mathcal{V}',
\]
be defined as
\[
\langle \gamma _{\tau }^{\ast }\xi ,\mathbf{w}\rangle _V\equiv
\int_0^{T}\int_{\Gamma _C}\xi \cdot \mathbf{w}_{\tau }\,dS dt.
\]

Finally, we assume that $\mathbf{f}(\cdot ,\omega ) \in
L^2(0,T;V')$ and it includes the body force $\mathbf{\hat{f}}$
in $U$ and the traction $\mathbf{t}$ on $\Gamma _{N}$.

The abstract form of problem \eqref{a1.1}--\eqref{a1.6}, is as follows.
\smallskip

\noindent Problem $\mathbf{P}_{\rm abst}$.
Find $\mathbf{u},\mathbf{v}\in \mathcal{V}$ and
$\xi \in L^2(0,T;L^2(\Gamma _C) ^3)$ such that
\begin{equation}
\mathbf{v}'+M\mathbf{v}+A\mathbf{u}+P\mathbf{u}+\gamma _{\tau
}^{\ast }\xi =\mathbf{f}\quad \text{in }\mathcal{V}',
\label{a2.19}
\end{equation}
with $\mathbf{v}(0,\omega )=\mathbf{v}_0(\omega ) \in H$, and
\begin{equation}
\mathbf{u}(t,\omega )=\mathbf{u}_0(\omega ) +\int_0^{t}
\mathbf{v}(s,\omega )ds,\quad \mathbf{u}_0(\omega ) \in V,
\label{a2.21}
\end{equation}
and for all $\mathbf{w\in }\mathcal{V}$,
\begin{equation}
\langle \gamma _{\tau }^{\ast }\xi ,\mathbf{w}\rangle _V\leq
\int_0^{T}\int_{\Gamma _C}F(u_n-g)\mu (| \mathbf{v}
_{\tau }-\mathbf{v}^{\ast }| ) \cdot [
| \mathbf{v}_{\tau }-\mathbf{v}^{\ast }+\mathbf{w}_{\tau
}| -| \mathbf{v}_{\tau }-\mathbf{v}^{\ast
}| ] \,dS\,dt. \label{16jane9f}
\end{equation}

We note that if $\mathbf{v}$ solves the abstract problem $\mathbf{P}_{\rm abst}$,
then $\mathbf{u}$ is a weak solution of \eqref{a1.1}--\eqref{a1.6}. As
usual, other variational and stable boundary conditions can be incorporated
by the appropriate choice of $V$ and $\mathbf{f}(\cdot ,\omega )
\in L^2(0,T;V')$. The following is the main result for the cases
with continuous friction coefficient.

\begin{theorem}\label{thm16}
Let $\mathbf{u}_0(\omega ) \in V,\mathbf{v}
_0(\omega ) \in H$, for each $\omega \in \Omega $, these
functions being $\mathcal{F}$ measurable. Assume that
$\mathbf{f}(\cdot ,\omega ) \in \mathcal{V}'$, and the gap
$(t,\omega ) \to g(t,\omega )$ and the sliding velocity
$(t,\omega ) \to $ $\mathbf{v}^{\ast }(t,\omega )$
are $\mathcal{B}([ 0,T] ) \times \mathcal{F}$
measurable and bounded. Then, there exists a solution $(\mathbf{u},\mathbf{v}
)$ to the problem \eqref{a2.19}--\eqref{16jane9f} for each $\omega $. This
solution $(t,\omega ) \to (\mathbf{u}(t,\omega) ,\mathbf{v}(t,\omega ) )$ is
 measurable into $V,H$ and $V'$. If, in addition, the friction coefficient
$\mu $ is Lipschitz continuous, then the solution is unique.
\end{theorem}

To carry out the proofs of existence and uniqueness, we note that
both $M$ and $A$ are coercive, nonnegative, and symmetric. That is, for two
constants $\delta >0$, $\lambda \geq 0$ they satisfy the following conditions
\begin{equation}
\langle B\mathbf{u},\mathbf{u}\rangle \geq \delta ^2\|\mathbf{u}
\|_{W}^2-\lambda |\mathbf{u}|_{H}^2,\quad 
\langle B\mathbf{u},\mathbf{u} \rangle \geq 0,\quad
\langle B\mathbf{u},\mathbf{v}\rangle =\langle B\mathbf{v},\mathbf{u}\rangle , 
\label{a2.23}
\end{equation}
for $B=M$ or $A$. Indeed, \eqref{a2.23} is a consequence of
\eqref{a1.11}--\eqref{a1.13} and Korn's inequality \cite{Ole92}.

\subsection{An Approximate Problem}
\label{sec:approx}

To establish the theorem, we use a sequence of approximate problems that we
solve using the Galerkin method. To that end, we first regularize the
friction condition, which has a subgradient form. We approximate the norm
function $\gamma (\mathbf{r}) =| \mathbf{r}| $
with the function
\[
\Psi _{\varepsilon }(\mathbf{r}) =\sqrt{| \mathbf{r}
| ^2+\varepsilon },
\]
which is convex, Lipschitz continuous, and has bounded derivative,
and it converges uniformly to
$\gamma (\mathbf{r}) = | \mathbf{r}| $ on $\mathbb{R}$ as $\varepsilon \to 0$,
 moreover,
\[
| \Psi _{\varepsilon }(\mathbf{x}) - \Psi _{\varepsilon
}(\mathbf{y}) | \leq | \mathbf{x}-\mathbf{y}|, \quad
| \Psi _{\varepsilon }'(\mathbf{t } ) | \leq 1.
\]
Furthermore, $\Psi _{\varepsilon }'$ is Lipschitz continuous with
Lipschitz constant $C/\sqrt{\varepsilon }$, where $\Psi _{\varepsilon}'$
denotes the gradient or Frechet derivative of the scalar valued
function.

The approximate problem to which we apply the Galerkin method is obtained by
replacing the friction condition \eqref{16jane9f} with its regularization,
and is as follows.
\smallskip

\noindent Problem $\mathbf{P}_{\varepsilon }$.
Find $\mathbf{u},\mathbf{v} \in \mathcal{V}$ such that
\begin{equation}
\mathbf{v}'+M\mathbf{v}+A\mathbf{u}+P\mathbf{u}+\gamma _{\tau
}^{\ast }F(u_n-g(\omega )) \mu (| \mathbf{v}
_{\tau }-\mathbf{v}^{\ast }(\omega )| ) \Psi
_{\varepsilon }'(\mathbf{v}_{\tau }-\mathbf{v}
^{\ast }(\omega )) =\mathbf{f}
\label{11jan1}
\end{equation}
in $\mathcal{V}'$,
with $\mathbf{v}(0)=\mathbf{v}_0\in H$, where
\begin{equation}
\mathbf{u}(t)=\mathbf{u}_0+\int_0^{t}\mathbf{v}(s)ds,\quad \mathbf{u}
_0\in V. \label{11jan3}
\end{equation}
Here, we indicate that the gap and the velocity of the foundation are random
variables depending on $\omega \in \Omega $, which is fixed, and the
approximate friction operator is defined for $\mathbf{w}\in V$ in the
following manner,
\begin{align*}
&\langle \gamma _{\tau }^{\ast }F(u_n-g(\omega )) \mu
(| \mathbf{v}_{\tau }-\mathbf{v}^{\ast }(\omega
)| ) \Psi _{\varepsilon }'(\mathbf{v}_{\tau }-
\mathbf{v}^{\ast }(\omega )) ,\mathbf{w}\rangle \\
& =\int_{\Gamma _C}F(u_n-g(\omega )) \mu (
| \mathbf{v}_{\tau }-\mathbf{v}^{\ast }(\omega
)| ) \Psi _{\varepsilon }'(\mathbf{v}_{\tau }-
\mathbf{v}^{\ast }(\omega )) \cdot \mathbf{w}_{\tau }\,dS.
\end{align*}

Let $R$ denote the Riesz map from $V$ to $V'$ defined by $
\langle R\mathbf{u},\mathbf{v}\rangle_V =(\mathbf{u,v}
) _{H}$. Then, $R^{-1}:H\to V$ is a compact and self-adjoint
operator and so there exists a complete orthonormal basis $\{ \mathbf{e}
_k\}$ for $H$, such that $\{ \mathbf{e}_k\} \subseteq V$
and
\[
R\mathbf{e}_k=\lambda _k\mathbf{e}_k,
\]
where $\lambda _k\to \infty $. Let $V_n=\operatorname{span}\{ \mathbf{e}
_1,\dots ,\mathbf{e}_n\}$. Then, $\cup _nV_n$ is dense in $H$ and
is also dense in $V$, and $\{ \mathbf{e}_k\} $ is an orthogonal
set in $V$. Indeed, we have
\[
0=(\mathbf{e}_k,\mathbf{e}_{l}) _{H}=\frac{1}{\lambda _k}
(R\mathbf{e}_k,\mathbf{e}_{l}) _{H}=\frac{1}{\lambda _k}
\langle R\mathbf{e}_k,\mathbf{e}_{l}\rangle =\frac{1}{\lambda
_k}(\mathbf{e}_{l},\mathbf{e}_k) _V,\quad k\neq l.
\]
Next, to show that $\cup _nV_n$ is dense in $V$, assume that this is not
so, then there exists $f\in V',\ f \neq 0$, such that $\cup
_nV_n $ is in $\ker (f) $. But $f=R\mathbf{u}$, for some $
\mathbf{u}$, and so
\[
0=\langle R\mathbf{u,e}_k\rangle =\langle R\mathbf{e}_k,
\mathbf{u}\rangle_V =\lambda _k(\mathbf{e}_k,\mathbf{u}
) _{H},
\]
for all $\mathbf{e}_k$ and so $\mathbf{u=0}$ by the density of $\cup
_nV_n$ in $H$, and hence $R\mathbf{u}=0=f$ after all, a contradiction.

Now, we apply the Galerkin method to Problem $\mathbf{P}_{\varepsilon }$.
Let
\[
\mathbf{v}_k(t,\omega ) =\sum_{j=1}^{k}x_j(t,\omega) \mathbf{e}_j,\quad
\mathbf{u}_k(t,\omega )=\mathbf{u}_0+\int_0^{t}\mathbf{v}_k(s,\omega )ds,
\]
and let $\mathbf{v}_k$ be the solution to the following integral equation,
for each $\omega $ and $j\leq k$. We now suppress the dependence on $\omega $
to simplify the notation, unless it is needed.
\begin{equation}
\begin{aligned}
&\Big\langle \mathbf{v}_k(t) -\mathbf{v}_{0k}+\int_0^{t}M\mathbf{v}_k+A
\mathbf{u}_k+P\mathbf{u}_k\\
&+\gamma _{\tau }^{\ast }F(
u_{kn}-g) \mu (| \mathbf{v}_{k\tau }-\mathbf{v}^{\ast
}| ) \Psi _{\varepsilon }'(\mathbf{v}_{k\tau
}-\mathbf{v}^{\ast })
,\mathbf{e}_j\Big\rangle \\
&=\int_0^{t}\langle \mathbf{f,e} _j\rangle ds.
\end{aligned} \label{11jan4}
\end{equation}
Here, $\mathbf{v}_{0k}\to \mathbf{v}_0\in H$ and the equation
holds for each $\mathbf{e}_j$ for each $j\leq k$. Then, this integral
equation reduces to a system of ordinary differential equations for the
vector $\mathbf{x}(t,\omega )$ whose $j^{th}$ component is $
x_j(t,\omega )$ mentioned above. We will obtain existence and
measurability of $\mathbf{x}$ from Theorem \ref{thm11}.

We differentiate, multiply by $x_j$, add and then integrate and after some
manipulations we obtain various terms that need to be estimated. For the
friction term we have,
\begin{align*}
&\int_0^{t}\int_{\Gamma _C}F(u_{kn}-g) \mu (|
\mathbf{v}_{k\tau }-\mathbf{v}^{\ast }| ) \Psi
_{\varepsilon }'(\mathbf{v}_{k\tau }-\mathbf{v}
^{\ast }) \cdot \mathbf{v}_{k\tau }\,dS\,ds
\\
&=\int_0^{t}\int_{\Gamma _C}F(u_{kn}-g) \mu (
| \mathbf{v}_{k\tau }-\mathbf{v}^{\ast }|
) \Psi _{\varepsilon }'(\mathbf{v}_{k\tau }-
\mathbf{v}^{\ast }) \cdot (\mathbf{v}_{k\tau }-
\mathbf{v}^{\ast }) \,dS\,ds \\
&\quad +\int_0^{t}\int_{\Gamma _C}F(u_{kn}-g) \mu (
| \mathbf{v}_{k\tau }-\mathbf{v}^{\ast }|
) \Psi _{\varepsilon }'(\mathbf{v}_{k\tau }-
\mathbf{v}^{\ast }) \cdot \mathbf{v}^{\ast}\,dS\,ds.
\end{align*}
The first term on the right-hand side is nonnegative and the second term is
bounded below by an expression of the form
\begin{align*}
-C\int_0^{t}\int_{\Gamma _C}(1+| u_{kn}|
) | \mathbf{v}^{\ast }| \,dS\,ds
&\geq -C\int_0^{t}\| \mathbf{u}_k\| _{W}\|
\mathbf{v}^{\ast }\| _{L^2(\Gamma _C)^3}ds-C\\
&\geq -C\int_0^{t}\| \mathbf{u}_k\| _{W}-C,
\end{align*}
where $C$ is independent of $\varepsilon ,\omega $ and $k$.

Here, the space $W$ embeds compactly into $V$ and the trace map from
$W$ to $L^2(\Gamma _C) ^3$ is continuous. We note that the use of
the space $W$ is not essential here, however, below we do need this
intermediate space. To estimate the term with $P$, one uses the linear
growth condition of $P$ in \eqref{a1.8}.

It follows from equivalence of norms in finite dimensional spaces, the
assumed estimates on $M$, $A$, and $P$, and standard manipulations depending
on compact embeddings, that there exists an estimate suitable to apply
Theorem \ref{thm11} to obtain the existence of a solution such that $(
t,\omega ) \to \mathbf{x}(t,\omega )$ is
measurable into $\mathbb{R}^{k}$ which implies that $(t,\omega )
\to \mathbf{v}_k(t,\omega )$ is product measurable
into $V$ and $H$. This yields the measurable Galerkin approximation of a
solution.

Also, the estimates and compact embedding results for Sobolev spaces lead to
the inequality
\begin{equation}
| \mathbf{v}_k(t) |
_{H}^2+\int_0^{T}\| \mathbf{v}_k\|
_V^2ds+\| \mathbf{u}_k(t) \| _V^2\leq
C, \label{11jan5}
\end{equation}
where the constant $C$ does not depend on $\varepsilon $ or $k$.

Next, we need to estimate the time derivative in $\mathcal{V}'$.
The integral equation implies that for all $\mathbf{w}\in V_k$,
\begin{equation}
\begin{aligned}
&\langle \mathbf{v}_k'(t) ,\mathbf{w} \rangle _V\\
&+\langle M\mathbf{v}_k+A\mathbf{u}_k+P\mathbf{u}_k+\gamma _{\tau }^{\ast
}F(u_{kn}-g) \mu (| \mathbf{v}_{k\tau }-\mathbf{v}^{\ast
}| ) \Psi _{\varepsilon }'(\mathbf{v}_{k\tau }-\mathbf{v}^{\ast })
,\mathbf{w}\rangle _V\\
&=\langle \mathbf{f,w}\rangle ,
\end{aligned}\label{16jane1f}
\end{equation}
where the dependence on $t$ and $\omega $ is suppressed. In terms of inner
products in $V$ this reduces to
\begin{align*}
&(R^{-1}\mathbf{v}_k'(t) ,\mathbf{w})_V\\
&+\Big(R^{-1}(M\mathbf{v}_k+A\mathbf{u}_k+P\mathbf{u}_k+\gamma _{\tau }^{\ast
}F(u_n-g) \mu (| \mathbf{v}_{k\tau }-\mathbf{v}^{\ast
}| ) \Psi _{\varepsilon }'(\mathbf{v}_{k\tau }-\mathbf{v}^{\ast })
) ,\mathbf{w}\Big) _V\\
&=(R^{-1}\mathbf{f},\mathbf{w}) _V.
\end{align*}
In terms of the orthogonal projection in $V$ onto $V_k$, denoted by $P_k$,
this takes the form
\begin{align*}
&(R^{-1}\mathbf{v}_k'(t) ,P_k\mathbf{w})_V\\
&+(R^{-1}\Big( M\mathbf{v}_k+A\mathbf{u}_k+P\mathbf{u}_k+\gamma _{\tau }^{\ast
}F(u_n-g) \mu (| \mathbf{v}_{k\tau }-\mathbf{v}^{\ast
}| ) \Psi _{\varepsilon }'(\mathbf{v}_{k\tau
}-\mathbf{v}^{\ast })
\Big) ,P_k\mathbf{w}) _V
\\
&=(R^{-1}\mathbf{f},P_k\mathbf{w}) _V,
\end{align*}
for all $\mathbf{w}\in V$. Now $\mathbf{v}_k'(t) \in V_k\ $and so the first
term can be simplified and we can write
\begin{align*}
&(R^{-1}\mathbf{v}_k'(t) ,\mathbf{w})_V\\
&+(R^{-1}\Big( M\mathbf{v}_k+A\mathbf{u}_k+P\mathbf{u}_k+\gamma _{\tau }^{\ast
}F(u_n-g) \mu (| \mathbf{v}_{k\tau }-\mathbf{v}^{\ast
}| ) \Psi _{\varepsilon }'(\mathbf{v}_{k\tau
}-\mathbf{v}^{\ast })
\Big) ,P_k\mathbf{w}) _V
\\
&=(R^{-1}\mathbf{f},P_k\mathbf{w}) _V,
\end{align*}
for all $\mathbf{w}\in V$. Then it follows that for all $\mathbf{w}\in V$,
\begin{align*}
&(R^{-1}\mathbf{v}_k'(t) ,\mathbf{w})_V\\
&+(P_kR^{-1}\Big(
M\mathbf{v}_k+A\mathbf{u}_k+P\mathbf{u}_k+\gamma _{\tau }^{\ast
}F(u_n-g) \mu (| \mathbf{v}_{k\tau }-\mathbf{v}^{\ast
}| ) \Psi _{\varepsilon }'(\mathbf{v}_{k\tau
}-\mathbf{v}^{\ast })
\Big) ,\mathbf{w}) _V \\
&=(P_kR^{-1}\mathbf{f},\mathbf{w}) _V.
\end{align*}
Thus, in $V$ we have
\begin{align*}
&R^{-1}\mathbf{v}_k'(t)
+P_kR^{-1}\big(
M\mathbf{v}_k+A\mathbf{u}_k+P\mathbf{u}_k
+\gamma _{\tau }^{\ast}F(u_n-g) \mu (| \mathbf{v}_{k\tau }
-\mathbf{v}^{\ast }| ) \Psi _{\varepsilon }'(\mathbf{v}_{k\tau
}-\mathbf{v}^{\ast })\big) \\
&=P_kR^{-1}\mathbf{f}.
\end{align*}
and $R^{-1}$ preserves norms while $P_k$ decreases them. Hence, the
estimate \eqref{11jan5} implies that $\| \mathbf{v}_k'\| _{\mathcal{V}'}$
is also bounded independently
of $\varepsilon $ and $k$. Then, summarizing the above estimates and
restoring $\omega $, yields
\begin{equation}
| \mathbf{v}_k(t,\omega ) | _{H}+\|
\mathbf{v}_k(\cdot ,\omega ) \| _{\mathcal{V}
}+\| \mathbf{v}_k'(\cdot ,\omega )
\| _{\mathcal{V}'}+\| \mathbf{u}_k(
t,\omega ) \| _V\leq C, \label{11jan6}
\end{equation}
where $C$ is a constant that does not depend on $\varepsilon $ and $k$.
Also, integrating \eqref{16jane1f}, leads to
\begin{equation}
\begin{aligned}
&i_k^{\ast }\Big(\mathbf{v}_k(t) -\mathbf{v}
_{0k}+\int_0^{t}M\mathbf{v}_kds+\int_0^{t}A\mathbf{u}
_kds+\int_0^{t}P\mathbf{u}_k\,ds
\\
&+\int_0^{t}\gamma _{\tau }^{\ast }F(u_{kn}-g) \mu
(| \mathbf{v}_{k\tau }-\mathbf{v}^{\ast
}| ) \Psi _{\varepsilon }'(\mathbf{v}
_{k\tau }-\mathbf{v}^{\ast }) ds\Big) \\
&=i_k^{\ast }\int_0^{t}\mathbf{f}ds,
\end{aligned} \label{16jane3f}
\end{equation}
where $i_k^{\ast }$ is the dual map to the inclusion map
$i_k:V_k\to V$.

Let $W$ be an intermediate space introduced above such that
\[
V\subseteq W\subseteq H,\quad V\text{ dense in }W,
\]
where the embedding is compact and the trace map onto $L^2(U)$
is continuous. Using Theorems \ref{t2.3a} and \ref{t2.4}, it follows that
for each fixed $\omega \in \Omega $, the following convergences hold true
for suitable subsequences, still denoted as $\{ \mathbf{v}_k\} $
, which may depend on $\omega $. We note that the compactness of the
embedding of $V$ into $W$ and consequently into $H$ implies the compactness
of the embedding of $H=H'$ into $W'$. Then using the
estimates and these theorems, as $k\to \infty $, we obtain
\begin{gather}
\mathbf{v}_k\rightharpoonup \mathbf{v}\quad \text{in }\mathcal{V},
\label{16jane5f}
\\
\mathbf{v}_k'\rightharpoonup \mathbf{v}'\quad \text{in } \mathcal{V}',
\\
\mathbf{v}_k\to \mathbf{v}\quad \text{strongly in }C([ 0,T] ,W') ,
\\
\mathbf{v}_k\to \mathbf{v}\quad \text{strongly in }L^2([0,T] ;W) ,
\\
\mathbf{v}_k(t) \to \mathbf{v}(t) \quad \text{in $W$ for a.a. }t,
\\
\mathbf{u}_k\to \mathbf{u}\quad \text{strongly in }C([ 0,T] ;W) ,
\\
A\mathbf{u}_k\rightharpoonup A\mathbf{u}\quad \text{in }\mathcal{V}',
\\
M\mathbf{v}_k\rightharpoonup M\mathbf{v}\quad \text{in }\mathcal{V}'.
\label{16jane6f}
\end{gather}

It follows from these convergences and the density of $\cup _nV_n$ in $V$
that by passing to a limit and using the dominated convergence theorem and
the strong convergences above in the nonlinear terms, we obtain the
following equation that holds in $V'$,
\begin{equation}
\begin{aligned}
&\mathbf{v}(t) +\int_0^{t}M\mathbf{v}ds+\int_0^{t}A\mathbf{u}
ds+\int_0^{t}P\mathbf{u}ds \\
&+\int_0^{t}\gamma _{\tau }^{\ast }F(u_n-g(\omega )) \mu
(| \mathbf{v}_{\tau }-\mathbf{v}^{\ast }(\omega
)| ) \Psi _{\varepsilon }'(\mathbf{v}_{\tau
}-\mathbf{v}^{\ast }(\omega )) ds\\
&=\mathbf{v} _0+\int_0^{t}\mathbf{f}ds.
\end{aligned} \label{16jan4f}
\end{equation}

Thus, $t\to \mathbf{v}(t,\omega )$ is continuous into $
V'$. This fact together with estimate \eqref{11jan6} imply that the
conditions of Theorem \ref{thm1} are satisfied. It follows that there is a
function $\mathbf{\bar{v}}$ which is product measurable into $V'$
and weakly continuous in $t$ and such that for each $\omega$ there is a
subsequence $\mathbf{v}_{k(\omega ) }$ such that $\mathbf{v}
_{k(\omega )}(\cdot ,\omega ) \rightharpoonup
\mathbf{\bar{v}}(\cdot,\omega )$ in $\mathcal{V}'$. By
repeating the above argument, for each $\omega$ we obtain that there exists
a further subsequence, still denoted as $\mathbf{v}_{k(\omega )
} $, that converges in $\mathcal{V}'$ to $\mathbf{v}(\cdot
,\omega )$, which is a solution \eqref{16jan4f} that is continuous
into $V'$. Hence, $\mathbf{\bar{v}}(\cdot ,\omega ) =
\mathbf{v }(\cdot ,\omega )$, and since these functions are
both weakly continuous into $V'$ they must be identical. Therefore,
there is a product measurable solution $\mathbf{v}$ to each regularized
problem.

It remains to pass to the regularization limit $\varepsilon \to 0$.
We let $\varepsilon =1/k$ and denote the product measurable solution of
\eqref{16jan4f} by $\mathbf{v}_k$ and note that estimate \eqref{11jan5} holds
true for $\mathbf{v}_k$. Then, we obtain a subsequence, still denoted as
$\mathbf{v}_k$, that has the same convergences as in
\eqref{16jane5f}--\eqref{16jane6f}. Thus, we obtain these convergences
along with the fact that $\mathbf{v}_k$ is product measurable and for
each $\omega $ it is a solution of the problem
\begin{equation}
\begin{aligned}
&\mathbf{v}_k(t) +\int_0^{t}M\mathbf{v}_k\,ds+\int_0^{t}A
\mathbf{u}_k\,ds+\int_0^{t}P\mathbf{u}_k\,ds\\
& +\int_0^{t}\gamma _{\tau }^{\ast }F(u_{kn}-g(\omega )
) \mu (| \mathbf{v}_{k\tau }-\mathbf{v}
^{\ast }(\omega )| ) \Psi _{1/k}'(\mathbf{v}
_{k\tau }-\mathbf{v}^{\ast }(\omega )) ds\\
&=\mathbf{v} _0+\int_0^{t}\mathbf{f}ds.
\end{aligned} \label{16jane7f}
\end{equation}
Next, in addition to \eqref{16jane5f}--\eqref{16jane6f}, we have
\[
\Psi _{1/k}'(\mathbf{v}_{k\tau }-\mathbf{v}^{\ast
}) \rightharpoonup \xi \quad \text{ in }L^{\infty }([ 0,T
] ;L^{\infty }(\Gamma _C) ^3) ,
\]
and moreover,
\[
\Psi _{1/k}'(\mathbf{v}_{k\tau }-\mathbf{v}^{\ast
}) \cdot \mathbf{w}_{\tau }\leq \Psi _{1/k}(\mathbf{v}_{k\tau }-
\mathbf{v}^{\ast }+\mathbf{w}_{\tau }) -\Psi _{1/k}(
\mathbf{v}_{k\tau }-\mathbf{v}^{\ast }) .
\]
Therefore, passing to the limit as $k\to \infty $; using the strong
convergence in the space $L^2([ 0,T] ;W) $, of $\mathbf{v}
_{k\tau }$ to $\mathbf{v}_{\tau }$; the uniform convergence of $\Psi _{1/k}$
to $\| \cdot \| $; the pointwise convergence in $W$; and
the dominated convergence theorem, we obtain that for $\mathbf{w}\in
\mathcal{V}$,
\begin{align*}
&\int_0^{t}\int_{\Gamma _C}F(u_{kn}-g(\omega )
) \mu (| \mathbf{v}_{k\tau }-\mathbf{v}
^{\ast }(\omega )| ) \Psi _{1/k}'(\mathbf{v}
_{k\tau }-\mathbf{v}^{\ast }(\omega )) \cdot \mathbf{w}
_{\tau }\,dS\,ds \\
&\quad \to \int_0^{t}\int_{\Gamma _C}F(
u_n-g(\omega ) ) \mu (| \mathbf{v}_{\tau
}-\mathbf{v}^{\ast }(\omega )| ) \xi \cdot
\mathbf{w}_{\tau }\,dS\,ds,
\end{align*}
where
\begin{equation}
\int_0^{t}\int_{\Gamma _C}\xi \cdot \mathbf{w}_{\tau }\,dS\,ds
\leq \int_0^{t}\int_{\Gamma _C}(| \mathbf{v}_{k\tau }-
\mathbf{v}^{\ast }+\mathbf{w}_{\tau }| -|
\mathbf{v}_{k\tau }-\mathbf{v}^{\ast }| ) \,dS\,ds.
\label{16jane8f}
\end{equation}
Then, passing to the limit in the integral equation \eqref{16jane7f}, we
obtain that for each $\omega $, $\mathbf{v}$ is a solution of the integral
equation
\begin{equation}
\begin{aligned}
&\mathbf{v}(t) +\int_0^{t}M\mathbf{v}\,ds+\int_0^{t}A\mathbf{u
}\,ds+\int_0^{t}P\mathbf{u}\,ds\\
&\quad +\int_0^{t}\gamma _{\tau }^{\ast }F(u_n-g(
\omega ) ) \mu (| \mathbf{v}_{\tau }-
\mathbf{v}^{\ast }(\omega )| ) \xi \,ds\\
&=\mathbf{v}_0+\int_0^{t}\mathbf{f}\,ds,
\end{aligned} \label{540}
\end{equation}
where $\xi $ satisfies the inequality \eqref{16jane8f}.
In particular, $\mathbf{v}$ is continuous into $V'$ and now, the conclusion of the
measurable selection theorem applies and yields the existence of a
measurable solution to \eqref{540} for each $\omega $. Taking a weak
derivative, it follows that we have obtained a product measurable solution
to the system \eqref{a2.19}--\eqref{16jane9f}. This completes the existence
part of the proof of Theorem \ref{thm16}. \vskip4pt

We note that when the friction coefficient $\mu(\cdot)$ is Lipschitz
continuous, one can show that for each $\omega $ the solution of the
integral equation \eqref{16jane7f} is unique, although this it is not an
obvious statement, see \cite{KS02}. This follows from standard procedures
involving Gronwall's inequality and the various necessary estimates.
Therefore, it is possible to obtain the product measurability by using more
elementary methods.

We also note that it allows one to include a stochastic
integral of the form $\int_0^{t}\Phi dW$. In this case one must consider a
filtration and obtain solutions that are adapted to the filtration.

In the next section we consider the case of discontinuous friction coefficient
and in this case it is not clear whether there is uniqueness, but we still
obtain a measurable solution.

\subsection{Discontinuous coefficient of friction}
\label{sec:discont}

In this section we consider the case when the coefficient of friction is a
discontinuous function of the slip speed, which was studied by us in \cite
{KS02}. This is the case described in elementary physics and engineering
courses, as well as in a host of engineering publications on friction, which
assumes, based on experimental data, that the coefficient of sliding or
dynamic friction is less than the coefficient of static friction. Additional
information can be found in \cite{SST04book} and the many references
therein. Therefore, we assume the friction coefficient function $\mu$ has a
jump discontinuity at 0, becoming smaller when the slip speed is positive.
The graph of the multi-function friction coefficient $\mu$ is depicted in
Figure \ref{fig2} in blue.

\begin{figure}[ht]
\begin{center}
\begin{picture}(230,150)(-20,-20)
\put(-20,0){\vector(1,0){230}}
\put(0,-20){\vector(0,1){150}}
\put(0,100){\qbezier(-2,0)(0,0)(2,0)}
{\color[rgb]{0,0,1} \put(0,100){\thicklines \line(0,-1){60}}
 \put(0,99.5){\thicklines \line(0,-1){60}}}
\put(-15,100){$\mu_0$}
\put(180,-15){$| \mathbf{v}-\mathbf{v}^{\ast}|$}
{\color[rgb]{0,0,1} \thicklines \qbezier(0,40)(100,5)(200,5)}
 \put(0,60){\qbezier(0,40)(2,-50)(200,-55) }
\put(90,7){$\mu_s$}
\put(-27,30){$\mu_s(0)$}
\put(17,60){$\mu_k$}
\end{picture}
\end{center}
\caption{Graph of $\protect\mu$ and $\protect\mu_k$ vs. the slip-rate $
|\mathbf{v}-\mathbf{v}^{\ast}|$.}
\label{fig2}
\end{figure}

We assume the friction coefficient is a set-valued function $\mu=\mu(r)$ that
consists of a Lipschitz continuous function $\mu _{s}$ and the segment
connecting the static friction coefficient $\mu_0$ and the value $
\mu_{s}(0)$ on the vertical axis, Figure \ref{fig2}.

To study the frictional contact problem with discontinuous friction
coefficient, we regularize the coefficient, obtain a measurable solution to
each regularized problem as above, and then pass to the limit. To that end,
we approximate by the multi-function $\mu $ with the sequence of functions $
\mu _k$, Figure \ref{fig2}, which are Lipschitz continuous and
converge uniformly to $\mu $ on every interval of the form $[\delta ,\infty )$ for $\delta >0$. It follows from Theorem \ref{thm16} that for each $k$
there exists a unique measurable solution of the integral equation
\begin{equation}
\begin{aligned}
&\mathbf{v}_k(t) +\int_0^{t}M\mathbf{v}_k\,ds+\int_0^{t}A
\mathbf{u}_k\,ds+\int_0^{t}P\mathbf{u}_k\,ds \\
&\quad +\int_0^{t}\gamma _{\tau }^{\ast }F(u_{kn}-g(\omega )
) \mu _k(| \mathbf{v}_{k\tau }-\mathbf{v}^{\ast }(\omega )| ) \xi _kds
=\mathbf{v}_0+\int_0^{t} \mathbf{f}\,ds,
\end{aligned} \label{17jane1f}
\end{equation}
where
\begin{equation}
\int_0^{t}\int_{\Gamma _C}\xi _k\cdot \mathbf{w}_{\tau }\,dS\,ds\leq
\int_0^{t}\int_{\Gamma _C}| \mathbf{v}_{k\tau }-
\mathbf{v}^{\ast }+\mathbf{w}_{\tau }| -| \mathbf{v}
_{k\tau }-\mathbf{v}^{\ast }| \,dS\,ds. \label{18jane2f}
\end{equation}

Let $\gamma (\mathbf{r}) =| \mathbf{r}| $,
then it follows from \eqref{18jane2f} that for $\omega $ off a set of
measure zero $\xi _k\in \partial \gamma (\mathbf{v}_{k\tau }-
\mathbf{v}^{\ast })$ a.e. $t$ for each $k$. Thus,
\begin{align*}
&\int_0^{t}\int_{\Gamma _C}F(u_{kn}-g) \mu _k(
| \mathbf{v}_{k\tau }-\mathbf{v}^{\ast }|
) \xi _k\cdot \mathbf{w}_{\tau }\,dS\,ds \\
&\leq \int_0^{t}\int_{\Gamma _C}F(u_{kn}-g) \mu _k(
| \mathbf{v}_{k\tau }-\mathbf{v}^{\ast }|
) (| \mathbf{v}_{k\tau }-\mathbf{v}^{\ast }+
\mathbf{w}_{\tau }| -| \mathbf{v}_{k\tau }-
\mathbf{v}^{\ast }| ) \,dS\,ds.
\end{align*}

Now, for each $\omega $, the estimate \eqref{11jan5} holds true, thus,
\[
| \mathbf{v}_k(t) |_{H}^2+\int_0^{T}\| \mathbf{v}_k\|
_V^2ds+\| \mathbf{u}_k(t) \| _V^2\leq C,
\]
where $C$ does not depend on $k$. Since $\xi _k$ is bounded, it follows
from \eqref{17jane1f} and this estimate, that $\mathbf{v}_k'$ is
bounded in $\mathcal{V}'$. So,
\begin{equation}
| \mathbf{v}_k(t) |
_{H}^2+\int_0^{T}\| \mathbf{v}_k\|
_V^2ds+\| \mathbf{u}_k(t) \|
_V^2+\| \mathbf{v}_k'\| _{\mathcal{V}
'}\leq C. \label{11auge4f}
\end{equation}
As was noted above, the constant $C$ is independent of $k$. Now, for fixed $
\omega $, there exists a subsequence, still denoted as $\{ \mathbf{v}
_k\} $ such that the convergences obtained in \eqref{16jane5f}--(\eqref
{16jane6f} hold. Thus, as $k\to \infty $,
\begin{gather}
\mathbf{v}_k\rightharpoonup \mathbf{v}\quad \text{in }\mathcal{V},
\label{16jane5ff} \\
\mathbf{v}_k'\rightharpoonup \mathbf{v}'\quad \text{in }\mathcal{V}', \\
\mathbf{v}_k\to \mathbf{v}\quad \text{strongly in }C([ 0,T] ,W') , \\
\mathbf{v}_k\to \mathbf{v}\quad \text{strongly in }L^2([0,T] ;W) , \\
\mathbf{v}_k(t) \to \mathbf{v}(t) \quad \text{in $W$ for a. a. } t, \label{9auge3f}\\
\mathbf{u}_k\to \mathbf{u}\quad \text{strongly in }C([ 0,T] ;W) , \label{11auge2f}\\
A\mathbf{u}_k\rightharpoonup A\mathbf{u}\quad \text{in }\mathcal{V}', \\
M\mathbf{v}_k\rightharpoonup M\mathbf{v}\quad \text{in }\mathcal{V}'. \label{16jane6ff}
\end{gather}

We note that more can be said if a further subsequence is taken. Indeed,
\[
m(t:\| \mathbf{v}_k(t) -\mathbf{v}(
t) \| _{W}\geq \lambda ) <\frac{1}{\lambda }
\int_0^{T}\| \mathbf{v}_k-\mathbf{v}\| _{W}^2ds,
\]
and so there exists a subsequence, still denoted by $\{\mathbf{v}_k\}$,
such that
\[
m(t:\| \mathbf{v}_k(t) -\mathbf{v}(t) \| _{W}\geq 2^{-k}) <2^{-k}.
\]
The Borel-Cantelli lemma implies that there exists a set of measure
zero $\mathcal{N}$ such that for $t$ not in this set,
\[
\| \mathbf{v}_k(t) -\mathbf{v}(t)\| _{W}<2^{-k},
\]
for all $k$ sufficiently large. Thus, for all $k$ large enough,
\[
\| \mathbf{v}_{k\tau }(t) -\mathbf{v}_{\tau }(
t) \| _{L^2(\Gamma _C) }<\frac{C}{2^{k}}.
\]
It now follows from the usual proof of the completeness of the space $L^2$
that for $t\notin \mathcal{N}$,
\begin{equation}
\mathbf{v}_{k\tau }(t,x) \to \mathbf{v}_{\tau }(
t,x) \quad \text{a.e. } x. \label{9auge4f}
\end{equation}
Passing to a further subsequence, if necessary, we may also assume that
\begin{gather*}
\mu _k(| \mathbf{v}_{k\tau }-\mathbf{v}^{\ast
}| ) \rightharpoonup \hat{\mu}\ \text{ weak}\ast \text{ in }
L^{\infty }([ 0,T] ;L^{\infty }(\Gamma _C)), \\
\mu _k(| \mathbf{v}_{k\tau }-\mathbf{v}^{\ast
}| ) \xi _k\rightharpoonup \Sigma \ \text{ weak}\ast \text{
in }L^{\infty }([ 0,T] ,L^{\infty }(\Gamma
_C) ^3) .
\end{gather*}
Next, for a given $\mathbf{w}\in \mathcal{V}$, we consider only
those $(t,x)$ for which convergence takes place in
\eqref{9auge4f}, and denote the set as
\[
S_0\equiv \{ (t,x) \notin \mathcal{M}:| \mathbf{v
}_{\tau }(t,x) -\mathbf{v}^{\ast }|
=0\} ,
\]
where $\mathcal{M}$ is the subset of $([ 0,T] \times \Gamma _C$
where convergence does not take place. Then, from the description of the $
\mu _k$, for $k$ large enough, $\mu _k(| \mathbf{v}
_{k\tau }(t,x) -\mathbf{v}^{\ast }| )
\in [ \mu _{s}(0) -\varepsilon ,\mu _0] $. Let $B$
be the set of all those $(t,x) \in S_0$ for which $\hat{\mu}
(t,x) >\mu _0$ and suppose it has positive measure. Then,
since $S$ is the surface measure on $\Gamma _C$ and $(m\times
S) (B) >0$, it follows from the above weak convergence
that
\begin{align*}
\mu _0(m\times S) (B)
&= \int_0^{T}\int_{\Gamma
_C}\mu _0\mathcal{X}_{B}(t,x) \,dS\,dt \\
&\geq \int_0^{T}\int_{\Gamma _C}\lim_{k\to \infty }\mu
_k(| \mathbf{v}_{k\tau }-\mathbf{v}^{\ast
}| ) \mathcal{X}_{B}(t,x) \,dS\,dt \\
&= \lim_{k\to \infty }\int_0^{T}\int_{\Gamma _C}\mu _k(
| \mathbf{v}_{k\tau }-\mathbf{v}^{\ast }|
) \mathcal{X}_{B}(t,x) \,dS\,dt \\
&= \int_0^{T}\int_{\Gamma _C}\hat{\mu}(t,x) \mathcal{X}
_{B}(t,x) \,dS\,dt>\mu _0(m\times S) (B) ,
\end{align*}
which is a contradiction. Similarly, if we assume that $B$ consists of those
$(t,x) \in S_0$ for which $\hat{\mu}(t,x) <\mu _{s}(0) -\varepsilon $,
one obtains a contradiction unless $(m\times S) (B) =0$.
It follows that for a.e. $(t,x) \in S_0$,
\[
\hat{\mu}(t,x) \in [ \mu _{s}(0) -\varepsilon,\mu _0] .
\]
Since $\varepsilon $ is arbitrary, it follows that
$\hat{\mu}( t,x) \in [ \mu _{s}(0) ,\mu _0] $ for
a.e. $(t,x) $. Now, let
\[
S_{+}\equiv \{ (t,x) \notin \mathcal{M}:| \mathbf{v
}_{\tau }(t,x) -\mathbf{v}^{\ast }|
>0\} .
\]
Then, by the convergence \eqref{9auge4f}, for a.e. $(t,x)$,
\[
\mu _k(| \mathbf{v}_{k\tau }(t,x) -
\mathbf{v}^{\ast }| ) \to \mu (
| \mathbf{v}_{\tau }(t,x) -\mathbf{v}^{\ast}| ) ,
\]
and so similar arguments show that
$\hat{\mu}(t,x) =\mu ( | \mathbf{v}_{\tau }(t,x) -\mathbf{v}^{\ast}| )$
for these $(t,x)$ as well. Thus $\hat{\mu}$ is in the graph of
$\mu (| \mathbf{v}_{\tau }(t,x) -\mathbf{v}^{\ast }| )$ off a set of
measure zero. Now, consider the friction term,
\begin{align*}
&\int_0^{T}\int_{\Gamma _C}F(u_{kn}-g) \mu _k(
| \mathbf{v}_{k\tau }-\mathbf{v}^{\ast }|
) \xi _k\cdot \mathbf{w}_{\tau }\,dS\,dt\\
&\leq \int_0^{T}\int_{\Gamma _C}F(u_{kn}-g) \mu _k(
| \mathbf{v}_{k\tau }-\mathbf{v}^{\ast }|
) (| \mathbf{v}_{k\tau }-\mathbf{v}^{\ast }+
\mathbf{w}_{\tau }| -| \mathbf{v}_{k\tau }-
\mathbf{v}^{\ast }| ) \,dS\,dt.
\end{align*}
Then the weak convergence and the strong convergence above imply
\begin{align*}
&\int_0^{T}\int_{\Gamma _C}F(u_n-g) \Sigma \cdot \mathbf{w}_{\tau }\,dS\,dt\\
&\leq \int_0^{T}\int_{\Gamma _C}F(u_n-g) \hat{\mu}(
| \mathbf{v}_{k\tau }-\mathbf{v}^{\ast }+\mathbf{w}
_{\tau }| -| \mathbf{v}_{k\tau }-\mathbf{v}
^{\ast }| ) \,dS\,dt.
\end{align*}
Here $\hat{\mu}(t,x) \in \mu (| \mathbf{v}_{\tau}(t,x) -\mathbf{v}^{\ast }| )$.
Define $\xi \equiv {\Sigma }/{\hat{\mu}}$, which is well defined since $\hat{
\mu}\neq 0$. Then, the above expression takes the form
\begin{align*}
&\int_0^{T}\int_{\Gamma _C}F(u_n-g) \hat{\mu}(\frac{
\Sigma }{\hat{\mu}}) \cdot \mathbf{w}_{\tau }\,dS\,dt\\
&=\int_0^{T}\int_{\Gamma _C}F(u_n-g) \hat{\mu}\xi
\cdot \mathbf{w}_{\tau }\,dS\,dt \\
&\leq \int_0^{T}\int_{\Gamma _C}F(u_n-g) \hat{\mu}(
| \mathbf{v}_{k\tau }-\mathbf{v}^{\ast }+\mathbf{w}
_{\tau }| -| \mathbf{v}_{k\tau }-\mathbf{v}^{\ast }| ) \,dS\,dt,
\end{align*}
where $\hat{\mu}(t,x) \in \mu (| \mathbf{v}_{\tau
}(t,x) -\mathbf{v}^{\ast }| ) $.

We now return to the approximate integral equation \eqref{17jane1f}. The
strong convergence \eqref{11auge2f} is sufficient to pass to the limit in
the term involving $P$. Therefore, collecting the above results establishes
the following Proposition.

\begin{proposition}\label{11augl1f}
For fixed $\omega $, there exist four functions
$(\mathbf{v,u,}\hat{\mu},\xi )$ such that $\mathbf{v}\in \mathcal{V}$,
$\mathbf{u} \in C([ 0,T] ,V) $, $\mathbf{v}'\in \mathcal{V}'$
\begin{gather}
\mathbf{u}(t) =\mathbf{u}_0+\int_0^{t}\mathbf{v}(s) ds, \label{11auge3f} \\
\hat{\mu}\in \mu (| \mathbf{v}_{\tau }(t,x) -
\mathbf{v}^{\ast }| ) \quad \text{ a.e. }(t,x) . \nonumber
\end{gather}
For all $\mathbf{w}\in \mathcal{V}$,
\begin{align*}
&\int_0^{T}\int_{\Gamma _C}F(u_n-g) \hat{\mu}\xi \cdot
\mathbf{w}_{\tau }\,dS\,dt \\
&\leq \int_0^{T}\int_{\Gamma _C}F(u_n-g) \hat{\mu}
(| \mathbf{v}_{k\tau }-\mathbf{v}^{\ast }+\mathbf{w
}_{\tau }| -| \mathbf{v}_{k\tau }-\mathbf{v}^{\ast }| ) \,dS\,dt,
\end{align*}
and
\begin{equation}
\begin{aligned}
&\mathbf{v}(t) +\int_0^{t}M\mathbf{v}ds+\int_0^{t}A\mathbf{u}
\,ds+\int_0^{t}P\mathbf{u}\,ds
+\int_0^{t}\gamma _{\tau }^{\ast }F(u_n-g) \hat{\mu}\xi ds\\
&= \mathbf{v}_0+\int_0^{t}\mathbf{f}\,ds.
\end{aligned} \label{11auge1f}
\end{equation}
\end{proposition}

Finally, the remaining issue is to show the existence of a measurable
solution. However, this follows in the same way as above from the measurable
selection theorem, Theorem \ref{thm1}. The reasoning as above shows that for
a fixed $\omega $ every sequence has a subsequence that converges to a
solution of the integral equation \eqref{11auge3f}--\eqref{11auge1f} that is
continuous into $V'$, this continuity follows directly from the
integral equation \eqref{11auge1f}. Moreover, one can obtain the estimate
\eqref{11auge4f} for all of the sequence $\mathbf{v}_k$ for $\omega $ off a
set of measure zero. Therefore, Theorem \ref{thm1} asserts that there is a
function $\mathbf{v}(\cdot ,\omega )$ in $\mathcal{V}'$
that is product measurable into $V'$ that is also weakly continuous
in $t$ into $V'$, and there is a subsequence {$\mathbf{v}_{k(
\omega ) }(\cdot ,\omega )$ that converges weakly to $
\mathbf{v}(\cdot ,\omega )$ in $\mathcal{V}'$. We note
that although $\mathbf{v}$ has values in $V$, it is only known to be
continuous into $V'$, which has a weaker norm. Then, it follows
from the above argument that a further subsequence converges to a solution
of the integral equation, and since both are weakly continuous into $
V'$, this solution to the integral equation equals this measurable
function $\mathbf{v}$ for all $t$, and for each $\omega $ off a set of
measure zero. Thus, there is a measurable solution to the stochastic
friction problem with discontinuous friction coefficient, too. This result
is summarized in the following theorem. }

\begin{theorem}
For each $\omega \in \Omega$, let $\mathbf{u}_0(\omega ) \in V$,
$\mathbf{v}_0(\omega ) \in H$, and $\;\mathbf{f}(\cdot,\omega ) \in \mathcal{V}'$.
 Also, assume the gap $g$ and
sliding velocity $\mathbf{v}^{\ast }$ are $\mathcal{B}([ 0,T ] ) \times \mathcal{F}$
measurable, and $\mu $ has a jump discontinuity at the origin.
Then, there exists a solution $\mathbf{v}$
of the problem summarized in \eqref{11auge3f}--\eqref{11auge1f} for each
$ \omega \in \Omega $. This solution $(t,\omega ) \to\mathbf{v}(t,\omega )$
is product measurable into $V,H$ and $V'$.
\end{theorem}

It only remains to check the last claim about measurability into the spaces 
$V$ and $H$. By the density of $V$ into $H$, it follows that $H'$ is
dense in $V'$ and so a simple argument using the Pettis theorem
implies that $\omega \to \mathbf{v}(t,\omega )$ is
$\mathcal{F}$ measurable in both $V$ and $H$.

Finally, we note that if we assume more regularity on $\mathbf{f}$, say that
it is actually in $L^2([ 0,T] \times \Omega ;V')$, then we could say
that in fact, $\mathbf{v}\in L^2([0,T] \times \Omega ;V)$.
This is obtained by simply integrating
the estimate \eqref{11jan5} and being more careful about the structure of
the constant on the right-hand side in this inequality. The measurability
issue is obtained again from our major theorem. We have not done this
because we want to emphasize that this extra assumption is not needed in
order to get measurable solutions.


\subsection*{Acknowledgements}
We would like to thank the anonymous referee for the constructive comments
 that helped to improve this article.

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\end{document}