11institutetext: Boris S. Mordukhovich 22institutetext: Department of
Mathematics, Wayne State University, Detroit, Michigan 48202, USA
22email<EMAIL_ADDRESS>
33institutetext: Pedro Pérez-Aros 44institutetext: Instituto de Ciencias de la
Ingeniería, Universidad de O’Higgins, Rancagua, Chile
44email<EMAIL_ADDRESS>
#  Generalized Leibniz rules and Lipschitzian stability for expected-integral
mappings††thanks: Research of the first author was partially supported by the
USA National Science Foundation under grants DMS-1512846 and DMS-1808978, by
the USA Air Force Office of Scientific Research under grant #15RT04, and by
the Australian Research Council under Discovery Project DP-190100555. Research
of the second author was partially supported by ANID grant: Fondecyt Regular
1200283 and Fondecyt Regular 1190110.

Boris S. Mordukhovich  Pedro Pérez-Aros

###### Abstract

This paper is devoted to the study of the expected-integral multifunctions
given in the form

$\mathrm{E}_{\Phi}(x):=\int_{T}\Phi_{t}(x)d\mu,$

where $\Phi\colon T\times{\mathbb{R}^{n}}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ is a set-valued mapping on a
measure space $(T,\mathcal{A},\mu)$. Such multifunctions appear in
applications to stochastic programming, which require developing efficient
calculus rules of generalized differentiation. Major calculus rules are
developed in this paper for coderivatives of multifunctions
$\mathrm{E}_{\Phi}$ and second-order subdifferentials of the corresponding
expected-integral functionals with applications to constraint systems arising
in stochastic programming. The paper is self-contained with presenting in the
preliminaries some needed results on sequential first-order subdifferential
calculus of expected-integral functionals taken from the first paper of this
series.

###### Keywords:

Stochastic programming Generalized differentiation Integral multifunctions
Leibniz rules Lipschitzian stability

###### MSC:

Primary: 49J53, 90C15, 90C34 Secondary: 49J52

##  1 Introduction

Stochastic programming has been highly recognized as an important area of
optimization theory with a variety of practical applications; see, e.g., the
book sdr  and the references therein. Although advanced methods of variational
analysis and generalized differentiation have been used in the study and
applications of stochastic programming (see, e.g., ah ; ap ; burke ; chp20 ;
dent-rusz ; hhp ; hr ; mp20 ; mor-sag18 ; mor-sag19  among other
publications), there is no comparison between the number of currently achieved
results in this vein for stochastic problems and the broad implementation of
the aforementioned methods in deterministic optimization.

The primal motivation for our study is to narrow the gap between deterministic
and stochastic applications of variational analysis and generalized
differentiation to optimization and related problems. The underlying feature
of stochastic problems is the presence of integration with respect to
probability measures over extended-real-valued variable integrands as well as
over vector-valued and set-valued mappings. Then efficient calculus rules of
their generalized differentiation are in strong demand.

In our preceding paper mp20a  we established various sequential versions of
the generalized Leibniz rule (subdifferentiation under the integral sign) in
terms of regular subgradients of the expected-integral functionals that are
defined by

$\mathrm{E}_{\varphi}(x,\mathpzc{y}):=\int_{T}\varphi_{t}\big{(}x,\mathpzc{y}(t)\big{)}d\mu,$
(1.1)

where $x\in\mathbb{R}^{n}$,
$\mathpzc{y}\in\textnormal{L}^{1}(T;\mathbb{R}^{m})$,
$\varphi_{t}(x,y):=\varphi(t,x,y)$, and $\varphi\colon
T\times\mathbb{R}^{n}\times\mathbb{R}^{m}\to\overline{\mathbb{R}}:=[-\infty,\infty]$
is an extended-real-valued function on a complete finite measure space
$(T,\mathcal{A},\mu)$, which is assumed below.

The goal of this paper is to proceed much further and to study expected-
integral multifunctions (set-valued mapping) given in the form

$\mathrm{E}_{\Phi}(x):=\int_{T}\Phi_{t}(x)d\mu,$ (1.2)

where $\Phi\colon T\times{\mathbb{R}^{n}}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ is a set-valued (in particular,
single-valued $\Phi\colon T\times{\mathbb{R}^{n}}\to\mathbb{R}^{m}$) mapping
defined on a measure space. The reader may consult the survey paper hess  for
an overview on the integration of random set-valued mappings and set-valued
probability theory. We also refer the reader to the recent publications burke
; dent-rusz  and the bibliographies therein for new developments and
applications. Our current work presented in this paper is mostly theoretical,
while we’ll discuss the aimed stochastic applications in the concluding
section below.

A natural extension of subdifferentials to the case of (single-valued and set-
valued) mappings is provided by coderivatives, and thus we focus here on
deriving coderivative versions of Leibniz’s rule for expected-integral
multifunctions of type (1.2). To the best of our knowledge, this has never
been done in the literature. In contrast to mp20a , our major results are
obtained in the exact/pointwise form, i.e., they are formulated exactly at the
points in question via the limiting constructions. Such results are clearly
more convenient for the theory and applications than sequential/fuzzy ones and
provide new rules even for extended-integral functionals (1.1) in comparison
with mp20a .

An important role in deriving pointwise coderivative Leibniz-type rules is
played by the new integrable quasi-Lipschitz property for set-valued random
normal integrands $\Phi_{t}(x)$ in (1.2). For deterministic multifunctions,
the introduced property is equivalent to the (Aubin) Lipschitz-like property
due to the (Mordukhovich) coderivative criterion of variational analysis (see
m93 ; m06 ; rw ), while these two Lipschitzian properties are essentially
different in stochastic frameworks. It happens, in particular, for spaces with
nonatomic measures, where the integrable quasi-Lipschitz property occurs to be
equivalent to the integrable counterpart of the (Hausdorff) locally
Lipschitzian property for random multifunctions. All of this leads us to
pleasable conclusions about Lipschitz stability of random feasible solution
mappings in problems of stochastic programming. They include, in particular,
efficient conditions on quantitative continuity of parametric sets of feasible
solutions to stochastic programs with inequality constraints.

Along with the study of expected-integral multifunctions (1.2) defined by
arbitrary integrand mappings $\Phi_{t}(x)$, we consider structural
multifunctions of the type

$\Phi_{t}(x)=F\big{(}t,g_{t}(x)\big{)}\;\mbox{ for all }\;x\in U\;\mbox{ and
a.e.\ }\;t\in T$ (1.3)

defined as compositions of set-valued mappings $F$ and single-valued mappings
$g$ between finite-dimensional spaces. It is shown that such multifunctions
(1.2) exhibit integrable Lipschitz stability in the case of convex outer
mappings $F$ and smooth inner mappings $g$ in (1.3). Then we introduce a new
class of integrable amenable set-valued compositions and use them to establish
chain rules of coderivative integration for the corresponding expected-
integral multifunctions. The obtained results are specified for the case where
set-valued mappings $F$ in (1.3) are given by inequality constraints, which is
a typical case for feasible solutions mappings in constrained stochastic
programming.

Finally, in this paper we establish, for the first time in the literature,
Leibniz-type rules for second-order subdifferentials of expected-integral
functionals given in the form

$\mathrm{E}_{\varphi}(x):=\int_{T}\varphi_{t}\big{(}x\big{)}d\mu,$ (1.4)

generated by extended-real-valued functions $\varphi\colon
T\times{\mathbb{R}^{n}}\to\overline{\mathbb{R}}$. The obtained results are
specified for the case where $\varphi$ in (1.4) is given as a maximum
function, which is important for applications to problems of stochastic
programming.

The rest of the paper is organized as follows. Section 2 recalls and discusses
the basic constructions of generalized differentiation in variational analysis
that are systematically employed in the subsequent material. In Section 3 we
briefly review some notions of measurability and integration for
multifunctions and then present the sequential Leibniz rules for
subdifferentiation of expected-integral functionals (1.1) taken from mp20  and
used in what follows.

Section 4 is devoted to the study of Lipschitzian properties of random
multifunctions and presents various characterizations of all the three
properties mentioned above with establishing relationships between them in
general measure spaces as well as in spaces with purely atomic and nonatomic
measures.

In Section 5 we obtain several results of the new type labeled as coderivative
Leibniz rules, which evaluate both regular and limiting coderivatives of
expected-integral multifunctions (1.2) via the integration of random
integrands $\Phi_{t}$ therein. The results obtained include sequential Leibniz
rules for regular coderivatives and pointwise ones for the limiting
coderivative construction. As a consequence of the coderivative Leibniz rules
of the latter type and the aforementioned coderivative characterizations of
Lipschitzian properties, we establish efficient conditions for Lipschitz
stability of the expected-integral multifunctions in terms of coderivatives of
their integrands $\Phi_{t}(x)$.

Section 6 addresses the setting where the integrand $\Phi_{t}(x)$ is given in
the composition form (1.3) defined by integrable amenable mappings. The
obtained conditions for Lipschitz stability and composite Leibniz rules are
specified here for random sets of feasible solutions in constrained stochastic
programming.

Considering in Section 7 expected functionals (1.4), we derive sequential and
pointwise second-order Leibniz rules in terms of second-order subdifferentials
of two types, which have been well known in variational analysis while have
never been used in the study of expected-integral functionals and applications
to stochastic programming. The concluding Section 8 summarizes the main
achievements of this paper and discusses some directions of our future
research and applications.

In this paper we use the standard notation from variational analysis,
generalized differentiation, and stochastic programming; see, e.g., m06 ; rw ;
sdr . Recall that the extended real line is denoted by
$\overline{\mathbb{R}}:=[-\infty,\infty]$ with the conventions that
$(\pm\infty)\cdot 0=0\cdot(\pm\infty)=0$ and
$\infty-\infty=-\infty+\infty=\infty$. The symbol $\mathbb{B}_{r}(x)$ stands
for the closed ball centered at $x$ with radius $r>0$, while the unit closed
ball of the space in question is denoted simply by $\mathbb{B}$. Given a
nonempty set $\Omega\subset{\mathbb{R}^{n}}$, its indicator function
$\delta_{\Omega}$ is defined by $\delta_{\Omega}(x):=0$ if $x\in\Omega$ and
$\delta_{\Omega}(x):=\infty$ otherwise, while the characteristic function
$\operatorname{\mathds{1}}_{\Omega}$ is defined by
$\operatorname{\mathds{1}}_{\Omega}(x):=1$ for $x\in\Omega$ and
$\operatorname{\mathds{1}}_{\Omega}(x):=0$ for $x\notin\Omega$. The symbol
$x\stackrel{{\scriptstyle\Omega}}{{\to}}\bar{x}$ means that $x\to\bar{x}$ with
$x\in\Omega$, and $\Omega^{c}$ denotes the complement of $\Omega$. Finally,
$\mathbb{N}:=\\{1,2,\ldots\\}$ and
$\mathbb{R}_{+}:=\\{\alpha\in\mathbb{R}\;|\;\alpha>0\\}$.

##  2 Preliminaries from Generalized Differentiation

In this section we recall some basic constructions of generalized
differentiation in variational analysis that are broadly used in what follows.

Let $\Phi\colon{\mathbb{R}^{n}}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ be a set-valued mapping with the
domain and graph given by, respectively,

$\mbox{\rm
dom}\,\Phi:=\big{\\{}x\in\mathbb{R}^{n}\;\big{|}\;\Phi(x)\neq\emptyset\big{\\}}\;\mbox{
and
}\;\operatorname{gph}\Phi:=\big{\\{}(x,y)\in\mathbb{R}^{n}\times\mathbb{R}^{m}\;\big{|}\;y\in\Phi(x)\big{\\}}.$

The Painlevé-Kuratowski outer limit of $F$ as $x\to\bar{x}$ is defined by

$\mathop{{\rm Lim}\,{\rm
sup}}_{x\to\bar{x}}\Phi(x):=\big{\\{}v\in\mathbb{R}^{m}\big{|}\;\exists\,\mbox{
seqs. }\;x_{k}\to\bar{x},\;v_{k}\to v\;\mbox{ s.t.
}\;v_{k}\in\Phi(x_{k})\big{\\}}.$ (2.1)

The regular/Fréchet normal cone to $\Omega$ at $\bar{x}\in\Omega$ is

$\widehat{N}(\bar{x};\Omega):=\Big{\\{}x^{*}\in\mathbb{R}^{n}\;\Big{|}\;\limsup_{x\stackrel{{\scriptstyle\Omega}}{{\to}}\bar{x}}\frac{\langle
x^{*},x-\bar{x}\rangle}{\|x-\bar{x}\|}\leq 0\Big{\\}}$ (2.2)

with $\widehat{N}(\bar{x};\Omega):=\emptyset$ if $\bar{x}\notin\Omega$. The
limiting/Mordukhovich normal cone to $\Omega$ at $\bar{x}$ is defined via
(2.1) by

$N(\bar{x};\Omega):=\mathop{{\rm Lim}\,{\rm
sup}}_{x\to\bar{x}}\widehat{N}(x;\Omega).$ (2.3)

Note that, in contrast to (2.2), the limiting normal cone (2.3) and the
associated coderivative and subdifferential constructions (see below) are
robust, meaning that they are closed-graph multifunctions with respect to
perturbations of the initial points.

Based on the normal cones (2.2) and (2.3) to the graph of
$\Phi\colon{\mathbb{R}^{n}}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ at
$(\bar{x},\bar{y})\in\operatorname{gph}\Phi$, we define the corresponding
regular and basic/limiting coderivatives of $\Phi$ at $(\bar{x},\bar{y})$ for
all $y^{*}\in\mathbb{R}^{m}$ by, respectively,

$\widehat{D}^{*}\Phi(\bar{x},\bar{y})(y^{*}):=\big{\\{}x^{*}\in{\mathbb{R}^{n}}\;\big{|}\;(x^{*},-y^{*})\in\widehat{N}\big{(}(\bar{x},\bar{y});\operatorname{gph}\Phi\big{)}\big{\\}},$
(2.4)
$D^{*}\Phi(\bar{x},\bar{y})(y^{*}):=\big{\\{}x^{*}\in{\mathbb{R}^{n}}\;\big{|}\;(x^{*},-y^{*})\in
N\big{(}(\bar{x},\bar{y});\operatorname{gph}\Phi\big{)}\big{\\}},$ (2.5)

where we omit $\bar{y}$ in the coderivative notation of $\Phi$ if it is a
singleton $\\{\Phi(\bar{x})\\}$. If
$\Phi\colon{\mathbb{R}^{n}}\to\mathbb{R}^{m}$ is ${\cal C}^{1}$-smooth around
$\bar{x}$ (in fact, merely strictly differentiable at this point), then both
coderivatives above reduce to the adjoint (transpose) Jacobian matrix linearly
applied to any $y^{*}\in\mathbb{R}^{m}$:

$\widehat{D}^{*}\Phi(\bar{x})(y^{*})=D^{*}\Phi(\bar{x})(y^{*})=\big{\\{}\nabla\Phi(\bar{x})^{*}y^{*}\big{\\}}.$
(2.6)

In general, both coderivatives (2.4) and (2.5) are positively homogeneous
multifunctions, where (2.4) is always convex-valued, while (2.5) is not even
for very simple convex functions as, e.g., for $\Phi(x):=|x|$ at
$\bar{x}=0\in\mathbb{R}$. Nevertheless, the limiting coderivative (2.5),
together with the normal cone (2.3) and the associated first-order and second-
order subdifferentials of extended-real-valued functions presented below,
enjoy full pointwise calculus based on variational and extremal principles of
variational analysis; see the books m06 ; m18 ; rw  for more details and
references. Unfortunately, it is not the case for the corresponding regular
constructions, for which only “fuzzy” results are available. On the other
hand, it is convenient to have the following representation:

$D^{*}\Phi(\bar{x},\bar{y})(y^{\ast})=\mathop{{\rm Lim}\,{\rm
sup}}_{(x,y)\stackrel{{\scriptstyle\operatorname{gph}\Phi}}{{\to}}(\bar{x},\bar{y}),\,v^{*}\to
y^{*}}\widehat{D}^{*}\Phi(x,y)(v^{*})$ (2.7)

of (2.5) at $(\bar{x},\bar{y})$ as the outer limit of (2.4) at points nearby.

Next we consider an extended-real-valued function
$\varphi\colon\mathbb{R}^{n}\to\overline{\mathbb{R}}$ with its domain and
epigraph that are defined, respectively, by

$\mbox{\rm
dom}\,\varphi:=\big{\\{}x\in\mathbb{R}^{n}\;\big{|}\;\varphi(x)<\infty\big{\\}}\;\mbox{
and }\;\mbox{\rm
epi}\,\varphi:=\big{\\{}(x,\alpha)\in\mathbb{R}^{n+1}\;\big{|}\;\alpha\geq\varphi(x)\big{\\}}.$

The properness of $\varphi$, which we assume from now on, means that
$\mbox{\rm dom}\,\varphi\neq\emptyset$ and $\varphi(x)>-\infty$ for all
$x\in{\mathbb{R}^{n}}$. Applying the normal cones (2.2) and (2.3) to the
epigraph of $\varphi$ at $(\bar{x},\varphi(\bar{x}))$ with
$|\varphi(\bar{x})|\neq\infty$ gives us the corresponding (geometric)
definitions of the regular and limiting/basic subdifferentials

$\widehat{\partial}\varphi(\bar{x}):=\big{\\{}x^{*}\in\mathbb{R}^{n}\;\big{|}\;(x^{*},-1)\in\widehat{N}\big{(}(\bar{x},\varphi(\bar{x}));\mbox{\rm
epi}\,\varphi\big{)}\big{\\}},$ (2.8)
$\partial\varphi(\bar{x}):=\big{\\{}x^{*}\in\mathbb{R}^{n}\;\big{|}\;(x^{*},-1)\in
N\big{(}(\bar{x},\varphi(\bar{x}));\mbox{\rm epi}\,\varphi\big{)}\big{\\}},$
(2.9)

while the reader is referred to m06 ; m18 ; rw  for equivalent analytic
descriptions, various properties, and applications. We say that the function
$\varphi$ is lower regular at $\bar{x}$ if $\varphi$ is finite and
$\operatorname{\widehat{\partial}}\varphi(\bar{x})=\operatorname{\partial}\varphi(\bar{x})$.

In the finite-dimensional setting under consideration, the regular
subdifferential (2.8) admits the following useful representation:

$\widehat{\partial}\varphi(\bar{x})=\big{\\{}x^{\ast}\in{\mathbb{R}^{n}}\;\big{|}\;\langle
x^{\ast},w\rangle\leq d{\varphi}(x)(w)\;\text{ for all
}\;w\in{\mathbb{R}^{n}}\big{\\}}$ (2.10)

in terms of the (Dini-Hadamard) subderivative of $\varphi$ at $\bar{x}$ with
respect to the direction $w$ defined by

$d{\varphi}(\bar{x})(w)=\liminf\limits_{t\downarrow 0,\,u\to
w}\frac{\varphi(\bar{x}+tu)-\varphi(x)}{t}.$ (2.11)

Finally in this section, we recall two notions of second-order
subdifferentials of extended-real-valued functions that are obtained by the
scheme of m92  as coderivatives of subgradient mappings; see m06 ; m18  for
more details. Given $\varphi\colon{\mathbb{R}^{n}}\to\overline{\mathbb{R}}$
finite at $\bar{x}$ and $\bar{x}^{\ast}\in\partial\varphi(\bar{x})$, the basic
second-order subdifferential of $\varphi$ at $\bar{x}$ relative to
$\bar{x}^{\ast}$ is defined as the set-valued mapping
$\partial^{2}\varphi(\bar{x},\bar{x}^{\ast})\colon{\mathbb{R}^{n}}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;{\mathbb{R}^{n}}$ with the values

$\partial^{2}\varphi(\bar{x},\bar{x}^{\ast})(v^{\ast}):=\big{(}D^{\ast}\partial\varphi\big{)}(\bar{x},\bar{x}^{\ast})(v^{\ast})\;\mbox{
whenever }\;v^{\ast}\in{\mathbb{R}^{n}}.$ (2.12)

The combined second-order subdifferential of $\varphi$ at $\bar{x}$ with
respect to $\bar{x}^{\ast}$ is defined similarly to (2.12) by replacing the
basic coderivative (2.5) in (2.12) with the regular one (2.4) as

$\breve{\partial}^{2}\varphi(\bar{x},\bar{x}^{\ast})(v^{\ast}):=\big{(}\widehat{D}^{\ast}\partial\varphi\big{)}(\bar{x},\bar{x}^{\ast})(v^{\ast})\;\mbox{
for all }\;v^{\ast}\in{\mathbb{R}^{n}}.$ (2.13)

The indication of $\bar{x}^{*}$ is dropped in the notation of (2.12) and
(2.13) when $\partial\varphi(\bar{x})=\\{\nabla\varphi(\bar{x})\\}$. Note that
for ${\cal C}^{2}$-smooth functions $\varphi$ we have

$\partial^{2}\varphi(\bar{x})(v^{\ast})=\breve{\partial}^{2}\varphi(\bar{x})(v^{\ast})=\big{\\{}\nabla^{2}\varphi(\bar{x})v^{*}\big{\\}}\;\mbox{
for any }\;v^{*}\in{\mathbb{R}^{n}}$

via the (symmetric) Hessian matrix. Calculus rules for the second-order
subdifferentials and their computations for remarkable classes of functions
can be found in m06 ; m18 ; mr  and the references therein.

##  3 Subdifferentiation of Expected-Integral Functionals

In this section we review the needed notions of measurability and integration
for set-valued mappings and then present some results on sequential Leibniz
rules for expected-integral functionals obtained in mp20 .

Throughout the paper, $(T,\mathcal{A},\mu)$ is a complete finite measure space
as mentioned in Section 1. To avoid confusions, we use the special font (as,
e.g., $\mathpzc{v},\mathpzc{w},\mathpzc{x},\mathpzc{y},\mathpzc{z}$, etc.) to
denote vector functions defined on $T$. When $p\in[1,\infty]$, the notation
$\textnormal{L}^{p}({T},\mathbb{R}^{n})$ stands for the space of all the
(equivalence classes by the relation equal almost everywhere) measurable
functions $\mathpzc{x}$ such that the scalar function
$\|\mathpzc{x}(\cdot)\|^{p}$ is integrable for $p\in[1,\infty)$ and
essentially bounded for $p=\infty$. The norm in
$\textnormal{L}^{p}(T,\mathbb{R}^{n})$ is denoted by $\|\cdot\|_{p}$, and the
points in ${\mathbb{R}^{n}}$ are identified with constant functions in
$\textnormal{L}^{p}(T,{\mathbb{R}^{n}})$. Thus for $x\in{\mathbb{R}^{n}}$ and
$\mathpzc{x}\in\textnormal{L}^{p}(T,{\mathbb{R}^{n}})$ we have the expressions

$\displaystyle\|x-\mathpzc{x}\|_{p}$
$\displaystyle:=\left(\int_{T}\|x-\mathpzc{x}(t)\|^{p}d\mu\right)^{1/p}\;\mbox{
as }\;p\in[1,\infty),$ $\displaystyle\|x-\mathpzc{x}\|_{\infty}$
$\displaystyle:=\operatorname*{ess\,sup}\limits_{t\in T}\|x-\mathpzc{x}(t)\|.$

Recall that a set-valued mapping $F\colon T\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{n}$ is measurable if for every open
set $U\subset\mathbb{R}^{n}$ the inverse image $F^{-1}(U):=\\{t\in
T\;|\;F(t)\cap U\neq\emptyset\\}$ is measurable, i.e.,
$F^{-1}(U)\in\mathcal{A}$. The mapping $F$ is said to be graph measurable if
$\operatorname{gph}F\in\mathcal{A}\otimes\mathcal{B}(\mathbb{R}^{n})$, where
$\mathcal{B}(\mathbb{R}^{n})$ is the Borel $\sigma$-algebra, i.e., the
$\sigma$-algebra generated by open subsets of $\mathbb{R}^{n}$. It is easy to
see, due to the completeness of the measure space $(T,\mathcal{A},\mu)$, that
a multifunction $F$ with closed values is measurable if and only if it is
graph measurable. The Aumann integral of $F\colon T\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{n}$ over a measurable set
$S\in\mathcal{A}$ is defined by

$\int_{S}F(t)d\mu:=\bigg{\\{}\int_{S}\mathpzc{x}^{*}(t)d\mu\;\bigg{|}\;\mathpzc{x}^{*}\in{\textnormal{L}}^{1}(T,{\mathbb{R}^{n}})\textnormal{
and }\mathpzc{x}^{*}(t)\in F(t)\text{ a.e.}\bigg{\\}}.$ (3.1)

The fundamental Lyapunov convexity theorem says that if the measure $\mu$ is
nonatomic on $T$ (i.e., there is no $A\in{\cal A}$ such that $\mu(A)>0$ and
for any $B\subset A$ with $B\in{\cal A}$ and $\mu(B)<\mu(A)$ it follows that
$\mu(B)=0$), then the integral set in (3.1) is closed and convex in
$\mathbb{R}^{n}$ provided that the multifunction $F$ is measurable and
uniformly bounded by a summable on $T$ function. The reader is referred to the
book du  and its bibliography for the Lyapunov convexity theorem and its
infinite-dimensional extensions; see also mor-sag18  for the most recent
results in this direction.

Together with (3.1), in this paper we often use the following notion for
extended-real-valued functions of two variables. A function $\varphi\colon
T\times\mathbb{R}^{n}\to\overline{\mathbb{R}}$ is called a normal integrand if
the multifunction $t\to\mbox{\rm epi}\,\varphi_{t}$ is measurable with closed
values. By the completeness of $(T,\mathcal{A},\mu)$, this is equivalent to
saying that $\varphi$ is

$\mathcal{A}\otimes\mathcal{B}(\mathbb{R}^{n})$-measurable, and that for every
$t\in T$ the function $\varphi_{t}:=\varphi(t,\cdot)$ is lower semicontinuous
(l.s.c.); see, e.g., (rw, , Corollary 14.34). The normal integrand $\varphi$
is proper if the function $\varphi_{t}$ is proper for each $t\in T$. If
furthermore $\varphi_{t}$ is a convex for all $t\in T$ that $\varphi$ is
called to be a convex normal integrand.

Motivated by the above definition of normal integrands for extended-real-
valued functions, we introduce now its counterpart for set-valued mappings.

######  Definition 1 (set-valued normal integrands)

We say that a mapping $\Phi\colon T\times{\mathbb{R}^{n}}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ is a set-valued normal integrand
on a measure space $(T,\mathcal{A},\mu)$ if for all $t\in T$ the multifunction
$\Phi_{t}:=\Phi(t,\cdot)$ has closed graph and the graph of $\Phi$ belongs to
$\mathcal{A}\otimes\mathcal{B}({\mathbb{R}^{n}}\times\mathbb{R}^{m})$. If in
addition the set $\operatorname{gph}\Phi_{t}$ is convex for a.e. $t\in T$,
then we say that $\Phi$ is a set-valued convex normal integrand.

Based on our previous discussions and the completeness of the measure space
$(T,\mathcal{A},\mu)$, we conclude that Definition 1 of set-valued normal
integrands amounts to saying that $t\mapsto\operatorname{gph}\Phi_{t}$ is a
measurable multifunction with closed values.

Next we present some results on measurable multifunctions and normal
integrands that are broadly used in what follows. The first proposition
concerns graph measurability of subgradient mappings generated by extended-
real-valued normal integrands as well as coderivatives associated with set-
valued normal integrand mappings.

######  Proposition 1 (graph measurability of subgradient and coderivative
mappings)

Let $\varphi\colon T\times{\mathbb{R}^{n}}\to\overline{\mathbb{R}}$ be a
proper normal integrand, and $\Phi\colon T\times{\mathbb{R}^{n}}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ be a proper set-valued normal
integrand. Then the following multifunctions are graph measurable:

  1. (i)

$t\mapsto\operatorname{gph}\widehat{\partial}\varphi_{t}=\big{\\{}(x,x^{\ast})\in\mathbb{R}^{2n}\big{|}\;x^{\ast}\in\widehat{\partial}\varphi_{t}(x)\big{\\}}$.

  2. (ii)

$t\mapsto\operatorname{gph}\partial\varphi_{t}:=\big{\\{}(x,x^{\ast})\in\mathbb{R}^{2n}\;\big{|}\;x^{\ast}\in\partial\varphi_{t}(x)\big{\\}}$.

  3. (iii)

$t\mapsto\operatorname{gph}\widehat{D}^{\ast}\Phi_{t}:=\big{\\{}(x,y,x^{\ast},y^{\ast})\in\mathbb{R}^{2(n+m)}\big{|}\;x^{\ast}\in\widehat{D}^{\ast}\Phi_{t}(x,y)(y^{\ast})\big{\\}}$.

  4. (iv)

$t\mapsto\operatorname{gph}D^{\ast}\Phi_{t}:=\big{\\{}(x,y,x^{\ast},y^{\ast})\in\mathbb{R}^{2(n+m)}\;\big{|}\;x^{\ast}\in
D^{\ast}\Phi_{t}(x,y)(y^{\ast})\big{\\}}$.

Proof. Item (i) is proved in (mp20, , Theorem 3.2), item (ii) follows from
(i), but it can be also derived from (rw, , Theorems 14.26 and 14.60).
Finally, items (iii) and (iv) follow from (i) and (ii), respectively, by their
applications to the normal integrand
$(t,x,y)\mapsto\delta_{\operatorname{gph}\Phi_{t}}(x,y)$. $\hfill\square$

The following result is classical in the theory of measurable multifunctions.
It asserts that a graph measurable multifunction with nonempty, while not
necessarily closed, values admits a measurable (single-valued) selection; see,
e.g., the book (cv, , Theorem III.22) and its references.

######  Proposition 2 (measurable selections)

Let $F\colon T\;{\lower 1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;{\mathbb{R}^{n}}$ be a graph measurable
multifunction with nonempty values on a measurable space
$(T,\mathcal{A},\mu)$. Then there exists a measurable single-valued mapping
$\mathpzc{x}\colon T\to{\mathbb{R}^{n}}$ such that $\mathpzc{x}(t)\in F(t)$
for a.e. $t\in T$.

Finally in this section, we recall the notion of expected-integral functionals
introduced in our preceding paper mp20  and present two sequential Leibniz-
type rules in terms of regular subgradients that were established therein. Let
$\varphi\colon
T\times{\mathbb{R}^{n}}\times\mathbb{R}^{m}\to\overline{\mathbb{R}}$ be a
proper normal integrand such that there exists
$\nu\in\textnormal{L}^{1}(T,\mathbb{R}_{+})$ ensuring the uniform boundedness
from below condition

$\displaystyle\varphi_{t}(v,w)\geq-\nu(t)\;\text{ for all
}\;v\in{\mathbb{R}^{n}}\;\;w\in\mathbb{R}^{m}\text{ and all }t\in T.$ (3.2)

As defined in (1.1), the expected-integral functional
$\mathrm{E}_{\varphi}\colon{\mathbb{R}^{n}}\times\textnormal{L}^{1}(T,\mathbb{R}^{m})\to\overline{\mathbb{R}}$
generated by a normal integrand $\varphi$ is given by the formula

$\displaystyle\mathrm{E}_{\varphi}(x,\mathpzc{y}):=\int_{T}\varphi_{t}(x,\mathpzc{y}(t))d\mu.$

It is well known in measure theory (see, e.g., bog ) that in a finite measure
space $(T,\mathcal{A},\mu)$ there exist measurable disjoint sets $T_{pa}$ and
$T_{na}$ such that $\mu_{pa}(\cdot):=\mu(\cdot\cap T_{pa})$ is purely atomic
and $\mu_{na}(\cdot):=\mu(\cdot\cap T_{na})$ is nonatomic. Moreover, $T_{pa}$
is a countable union of disjoint atoms.

In the following results we assume that for a given point of interest
$\bar{x}$ there exists $\rho>0$ such that

$\varphi_{t}(v,\cdot)\;\text{ is convex for all
}\;v\in\mathbb{B}_{\rho}(\bar{x})\;\text{ and }\;t\in T_{na}.$ (3.3)

Now we are ready to present two sequential subdifferential Leibniz rules for
expected-integral functionals taken, respectively, from Theorem 5.2 and
Theorem 5.4 of our preceding paper mp20 . Note that, although these results
are not written in the expected Leibniz form, they certainly can be treated as
approximate versions. Furthermore, such results will lead us to the desired
generalized Leibniz rules by employing limiting procedures under appropriate
qualification conditions.

######  Theorem 3.1 (subdifferential Leibniz rule, I)

Let $\varphi$ be a proper normal integrand on $(T,\mathcal{A},\mu)$ satisfying
(3.2) and (3.3) around some $\bar{x}\in{\mathbb{R}^{n}}$. Pick any
$p,q\in(1,\infty)$ with $1/p+1/q=1$ and suppose that
$(\bar{x}^{*},\bar{\mathpzc{y}}^{\ast})\in\widehat{\partial}\mathrm{E}_{\varphi}(\bar{x},\bar{\mathpzc{y}})$
are such that the function

$t\mapsto\inf_{{\mathbb{R}^{n}}\times\mathbb{R}^{m}}\big{\\{}\varphi_{t}(\cdot,\cdot)-\langle\bar{\mathpzc{y}}^{\ast}(t),\cdot\rangle\big{\\}}$

is integrable on $T$. Then there exist sequences
$\\{x_{k}\\}\subset{\mathbb{R}^{n}}$,
$\\{\mathpzc{x}_{k}\\}\subset\textnormal{L}^{p}({T},{\mathbb{R}^{n}})$,
$\\{\mathpzc{x}_{k}^{*}\\}\subset{\textnormal{L}}^{q}({T},{\mathbb{R}^{n}})$,
$\\{\mathpzc{y}_{k}\\}\subset\textnormal{L}^{1}(T,\mathbb{R}^{m})$, and
$\\{\mathpzc{y}_{k}^{\ast}\\}\subset\textnormal{L}^{\infty}(T,\mathbb{R}^{m})$
satisfying the assertions:

  1. (i)

$\big{(}\mathpzc{x}_{k}^{*}(t),\mathpzc{y}_{k}^{\ast}(t)\big{)}\in\widehat{\partial}\varphi_{t}\big{(}\mathpzc{x}_{k}(t),\mathpzc{y}_{k}(t)\big{)}$
for a.e. and all $k\in\mathbb{N}$.

  2. (ii)

$\|\bar{x}-x_{k}\|\to 0$, $\|\bar{x}-\mathpzc{x}_{k}\|_{p}\to 0$, and
$\|\bar{\mathpzc{y}}-\mathpzc{y}_{k}\|_{1}\to 0$ as $k\to\infty$.

  3. (iii)

$\displaystyle\bigg{\|}\int_{T}\mathpzc{x}_{k}^{*}(t)d\mu-\bar{x}^{\ast}\bigg{\|}\to
0$ and $\|\mathpzc{x}_{k}^{*}\|_{q}\|\mathpzc{x}_{k}-x_{k}\|_{p}\to 0$ as
$k\to\infty$.

  4. (iv)

$\displaystyle\int_{T}\Big{|}\varphi_{t}\big{(}\mathpzc{x}_{k}(t),\mathpzc{y}_{k}(t)\big{)}-\varphi_{t}\big{(}\bar{x},\bar{\mathpzc{y}}(t)\big{)}\Big{|}d\mu\to
0$ as $k\to\infty$.

  5. (v)

$\|\mathpzc{y}_{k}^{\ast}-\bar{\mathpzc{y}}^{\ast}\|_{\infty}\to 0$ as
$k\to\infty$.

######  Theorem 3.2 (subdifferential Leibniz rule, II)

Let $\varphi$ be a proper normal integrand on $(T,\mathcal{A},\mu)$ satisfying
(3.2) and (3.3) at $\bar{x}\in{\mathbb{R}^{n}}$, and let
$(\bar{x}^{*},\bar{\mathpzc{y}}^{\ast})\in\widehat{\partial}\mathrm{E}_{\varphi}(\bar{x},\bar{\mathpzc{y}})$
be chosen so that the function

$t\mapsto\inf_{\mathbb{B}_{\hat{\rho}}(\bar{x})\times\mathbb{R}^{m}}\big{\\{}\varphi_{t}(\cdot,\cdot)-\langle\bar{\mathpzc{y}}^{\ast}(t),\cdot\rangle\big{\\}}$

is integrable on $T$ for some $\hat{\rho}>0$. Then there exist sequences
$\\{x_{k}\\}\subset{\mathbb{R}^{n}}$,
$\\{\mathpzc{x}_{k}\\}\subset\textnormal{L}^{\infty}({T},{\mathbb{R}^{n}})$,
$\\{\mathpzc{x}_{k}^{\ast}\\}\subset{\textnormal{L}}^{1}({T},{\mathbb{R}^{n}})$,
$\\{\mathpzc{y}_{k}\\}\subset\textnormal{L}^{1}(T,\mathbb{R}^{m})$, and
$\\{\mathpzc{y}_{k}^{\ast}\\}\subset\textnormal{L}^{\infty}(T,\mathbb{R}^{m})$
such that:

  1. (i)

$\big{(}\mathpzc{x}_{k}^{*}(t),\mathpzc{y}_{k}^{\ast}(t)\big{)}\in\widehat{\partial}\varphi_{t}\big{(}\mathpzc{x}_{k}(t),\mathpzc{y}_{k}(t)\big{)}$
for a.e. and all $k\in\mathbb{N}$.

  2. (ii)

$\|\bar{x}-x_{k}\|\to 0$, $\|\bar{x}-\mathpzc{x}_{k}\|_{\infty}\to 0$, and
$\|\bar{\mathpzc{y}}-\mathpzc{y}_{k}\|_{1}\to 0$ as $k\to\infty$.

  3. (iii)

$\displaystyle\bigg{\|}\int_{T}\mathpzc{x}_{k}^{*}(t)d\mu-\bar{x}^{\ast}\bigg{\|}\to
0$, and
$\displaystyle\int_{T}\Big{\|}\mathpzc{x}_{k}^{*}(t)\Big{\|}\cdot\Big{\|}\mathpzc{x}_{k}(t)-x_{k}\Big{\|}d\mu\to
0$ as $k\to\infty$.

  4. (iv)

$\displaystyle\int_{T}\Big{|}\varphi_{t}\big{(}\mathpzc{x}_{k}(t),\mathpzc{y}_{k}(t)\big{)}-\varphi_{t}(\bar{x},\bar{\mathpzc{y}}(t)\big{)}\Big{|}d\mu\to
0$ as $k\to\infty$.

  5. (v)

$\|\mathpzc{y}_{k}^{\ast}-\bar{\mathpzc{y}}^{\ast}\|_{\infty}\to 0$ as
$k\to\infty$.

##  4 Lipschitzian Properties of Random Multifunctions

This section is devoted to the study of new Lipschitzian properties for random
multifunctions defined on finite measure spaces. Our main attention is paid to
the three major properties of such multifunctions that we label as integrably
local Lipschitzian, integrably quasi-Lipschitzian, and integrably Lipschitz-
like ones. We reveal relationships between these properties in various classes
of measure spaces and compare them with the corresponding properties of
deterministic multifunctions. On the one hand, the new Lipschitzian properties
are important to establish stability of random sets of feasible solutions in
stochastic programming, but on the other hand they play a crucial role in
deriving calculus rules to evaluate coderivatives of expected-integral
multifunctions that is given in the subsequent sections.

Starting with the local Lipschitzian property of deterministic multifunctions,
recall that $\Phi\colon{\mathbb{R}^{n}}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ is (Hausdorff) locally
Lipschitzian around $\bar{x}\in\mbox{\rm dom}\,\Phi$ if there are numbers
$\eta>0$ and $\ell\geq 0$ such that

$\Phi(x)\subset\Phi(x^{\prime})+\ell\|x-x^{\prime}\|\mathbb{B}\;\text{ for all
}\;x,x^{\prime}\in\mathbb{B}_{\eta}(\bar{x}).$ (4.1)

Having (4.1) in mind, we now define its random version for set-valued normal
integrands on the general measure spaces under consideration.

######  Definition 2 (integrable local Lipschitzian property of random
multifunctions)

Let $\Phi\colon T\times{\mathbb{R}^{n}}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ be a set-valued normal integrand
on a complete finite measure space $(T,\mathcal{A},\mu)$, and let
$\bar{x}\in\mbox{\rm dom}\,\mathrm{E}_{\Phi}$ be given for $E_{\Phi}$ taken
from (1.2). We say that $\Phi$ is integrably locally Lipschitzian around
$\bar{x}$ if there exist $\eta>0$,
$\ell\in\textnormal{L}^{1}(T,\mathbb{R}_{+})$, and $\widehat{T}\in\mathcal{A}$
with $\mu(T\backslash\widehat{T})=0$ such that

$\Phi_{t}(x)\subset\Phi_{t}(x^{\prime})+\ell(t)\|x-x^{\prime}\|\mathbb{B}\;\text{
for all }\;t\in\widehat{T}\;\text{ and
}\;x,x^{\prime}\in\mathbb{B}_{\eta}(\bar{x}).$ (4.2)

Let us present useful characterizations of integrably locally Lipschitzian
multifunctions in terms of distance functions. Recall that the Pompeiu-
Hausdorff distance between sets $\Omega_{1},\Omega_{2}\subset{\mathbb{R}^{n}}$
is given by

$\displaystyle{\rm haus}(\Omega_{1},\Omega_{2})$
$\displaystyle:=\max\left\\{\sup\limits_{x\in\Omega_{1}}{\rm
dist}(x;\Omega_{2}),\sup\limits_{x\in\Omega_{2}}{\rm
dist}(x;\Omega_{1})\right\\}$ (4.3)
$\displaystyle=\sup\limits_{x\in{\mathbb{R}^{n}}}\big{|}{\rm
dist}(x;\Omega_{1})-{\rm dist}(x;\Omega_{2})\big{|},$

where ${\rm dist}(x;\Omega)$ is the standard distance function in
${\mathbb{R}^{n}}$ with the convention that $\mbox{\rm
dist}\,(x,\emptyset):=\infty$. Here are the equivalent descriptions of the
integrable local Lipschitzian property from (4.2).

######  Proposition 3 (distance descriptions of integrable local Lipschitzian
multifunctions)

Let $\Phi\colon T\times{\mathbb{R}^{n}}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ be a set-valued normal integrand,
and let $\bar{x}\in\mbox{\rm dom}\,\mathrm{E}_{\Phi}$ for $\mathrm{E}_{\Phi}$
taken from (1.2). Then the following assertions are equivalent:

  1. (i)

There exist $\widehat{T}\in\mathcal{A}$ with $\mu(T\backslash\widehat{T})=0$,
$\eta>0$, and $\ell\in\textnormal{L}^{1}(T,\mathbb{R}_{+})$ such that for all
$t\in\widehat{T}$ and all $x\in\mathbb{B}_{\eta}$ we have the inclusion in
(4.2).

  2. (ii)

There exist $\widehat{T}\in\mathcal{A}$ with $\mu(T\backslash\widehat{T})=0$,
$\eta>0$, and $\ell\in\textnormal{L}^{1}(T,\mathbb{R}_{+})$ such that

$\displaystyle{\rm
haus}\big{(}\Phi_{t}(x),\Phi_{t}(x^{\prime})\big{)}\leq\ell(t)\|x-x^{\prime}\|\;\text{
for all }\;x,x^{\prime}\in\mathbb{B}_{\eta}(\bar{x}),\text{ }t\in\widehat{T}.$
(4.4)

  3. (iii)

In the setting of (i) we have that the function $x\mapsto{\rm
dist}(y;\Phi_{t}(x))$ is $\ell(t)$-Lipschitz continuous on
$\mathbb{B}_{\eta}(\bar{x})$.

  4. (iv)

In the setting (ii) we have the estimate

${\rm dist}\big{(}y;\Phi_{t}(x)\big{)}\leq\ell(t){\rm
dist}\big{(}x;\Phi_{t}^{-1}(y)\cap\mathbb{B}_{\eta}(\bar{x})\big{)}\;\text{
for all }\;x\in\mathbb{B}_{\eta}(\bar{x}),\quad y\in\mathbb{R}^{m},\;\text{
and }\;t\in\widehat{T}.$ (4.5)

Proof. Note that assertion (i) means that $\Phi$ is integrably locally
Lipschitzian around $\bar{x}$ by Definition 2. The equivalence between (i) and
(ii) easily follows from (4.2) and definition (4.3) of the Pompeiu-Hausdorff
distance. The representation of the latter distance presented in (4.3) ensures
the equivalence between (ii) and (iii). To finish the proof, it remains
verifying the equivalence between (ii) and (iv). Observe first that we can
always suppose that $\mathbb{B}_{\eta}(\bar{x})\subset\mbox{\rm
dom}\,\Phi_{t}$ for all $t\in\widehat{T}$. Indeed, the choice of $\bar{x}$
tells us that $\bar{x}\in\mbox{\rm dom}\,\Phi_{t}$ holds for a.e. $t\in T$.
Hence it follows from (ii) that ${\rm haus}(\,\Phi_{t}(x),\Phi_{t}(\bar{x}))$
is finite for all $x\in\mathbb{B}_{\eta}(\bar{x})$ and $t\in\widehat{T}$,
which means that $\mathbb{B}_{\eta}(\bar{x})\subset\mbox{\rm dom}\,\Phi_{t}$
whenever $t\in\widehat{T}$. On the other hand, assuming (iv) gives us
$t\in\widehat{T}$, $y_{t}\in\Phi_{t}(\bar{x})$, and
$x\in\mathbb{B}_{\eta}(\bar{x})$ such that (4.5) yields ${\rm
dist}\big{(}y_{t};\Phi_{t}(x)\big{)}\leq\ell(t)\|x-\bar{x}\|$, which shows the
necessity of $\mathbb{B}_{\eta}(\bar{x})\subset\mbox{\rm dom}\,\Phi_{t}$. The
equivalence between (ii) and (iv) follows now from (dr, , Proposition 3C.1),
which therefore completes the proof of this proposition.

The following example illustrates how the the distance characterizations of
Proposition 3 allow us to easily check the fulfillment of the integrable local
Lipschitzian property of random multifunctions.

###### Example 1

$($checking the integrable locally Lipschitzian property of multifunctions$)$
Let $F\colon T\;{\lower 1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$, be an integrable bounded
measurable multifunction, i.e., there exists
$\lambda\in\textnormal{L}^{1}(T,\mathbb{R}_{+})$ such that
$F(t)\subset\lambda(t)\mathbb{B}$, and let $b\colon
T\times{\mathbb{R}^{n}}\to\mathbb{R}^{m}$ and $A\colon
T\times{\mathbb{R}^{n}}\to\mathbb{R}^{m\times m}$ be two measurable mappings.
Take $\bar{x}\in{\mathbb{R}^{n}}$ for which there are $\eta>0$,
$\ell_{1}\in\mathbb{R}_{+}$, and
$\ell_{2}\in\textnormal{L}^{1}(T,\mathbb{R}_{+})$ ensuring that

$\|A(t,x)-A(t,x^{\prime})\|\leq\ell_{1}\|x-x^{\prime}\|\;\mbox{ and
}\;\|b(t,x)-b(t,x^{\prime})\|\leq\ell_{2}(t)\|x-x^{\prime}\|$

for all $x,x^{\prime}\in\mathbb{B}_{\eta}(\bar{x}),\;t\in T$, and that
$A(t,x)$ is nonsingular for such $t,x$. Then the mapping
$\Phi_{t}(x):=A(t,x)F(t)+b(t,x)$ is integrably locally Lipschitzian around
$\bar{x}$. Indeed, it follows from (rw, , Example 9.32) that for a.e. $t\in T$
the modulus of Lipschitz continuity of $\Phi_{t}$ on
$\mathbb{B}_{\eta}(\bar{x})$ can be bounded by
$\lambda(t)\ell_{1}+\ell_{2}(t)$. Thus we conclude from the distance
characterizations (4.4) of Proposition 3 that the above multifunction $\Phi$
enjoys the claimed Lipschitzian property.

Our next goal is to establish coderivative characterizations of integrably
locally Lipschitzian and related Lipschitzian properties of random
multifunctions. Recall first that for deterministic set-valued mappings
$\Phi\colon{\mathbb{R}^{n}}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ the local Lipschitzian property
(4.1) can be written in form (4.2), where the measure space is a single atom.
The latter property is characterized by

$\sup\big{\\{}\|x^{*}\|\;\big{|}\;x^{*}\in
D^{*}\Phi(\bar{x},\bar{y})(y^{*})\big{\\}}\leq\ell\|y^{*}\|\;\mbox{ for all
}\;\bar{y}\in\Phi(\bar{x}),\;y^{*}\in\mathbb{R}^{m},$ (4.6)

provided that $\Phi$ is closed-graph and is uniformly bounded around this
point; see (m93, , Theorem 5.11). Due the robustness of the coderivative (2.5)
and its representation (2.7), the pointwise characterization in (4.6) can be
equivalently written in the following forms: there exist $\eta>0$ and
$\ell\geq 0$ such that

$\sup\big{\\{}\|x^{*}\|\;\big{|}\;x^{*}\in
D^{*}\Phi(x,y)(y^{*})\big{\\}}\leq\ell\|y^{*}\|\;\mbox{ and
}\;\sup\big{\\{}\|x^{*}\|\;\big{|}\;x^{*}\in\widehat{D}^{*}\Phi(x,y)(y^{*})\big{\\}}\leq\ell\|y^{*}\|$
(4.7)

for all $x\in\mathbb{B}_{\eta}(\bar{x})$, $y\in\Phi(x)$, and
$y^{*}\in\mathbb{R}^{m}$.

Observe further that a graphical localization of the locally Lipschitzian
property (4.1) of deterministic multifunctions
$\Phi\colon{\mathbb{R}^{n}}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ is known as the Lipschitz-like
(pseudo-Lipschitz, Aubin) property of $\Phi\colon{\mathbb{R}^{n}}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ around
$(\bar{x},\bar{y})\in\operatorname{gph}\Phi$ defined as: there exist $\eta>0$
and $\ell\geq 0$ such that

$\Phi(x)\cap\mathbb{B}_{\eta}(\bar{y})\subset\Phi(x^{\prime})+\ell\|x-x^{\prime}\|\mathbb{B}\;\text{
for all }\;x,x^{\prime}\in\mathbb{B}_{\eta}(\bar{x}).$ (4.8)

As well recognized (see, e.g., (m06, , Theorem 1.42)), the locally
Lipschitzian property of $\Phi$ around $\bar{x}\in\mbox{\rm dom}\,\Phi$ is
equivalent to the Lipschitz-like property of $\Phi$ around $(\bar{x},\bar{y})$
for all $\bar{y}\in\Phi(\bar{x})$, provided that $\Phi$ is closed-graph and
uniformly bounded around $\bar{x}$. Due to this fact, the above
characterization (4.6) of the locally Lipschitzian property of deterministic
multifunctions is a consequence of the following characterizations of the
Lipschitz-like property of $\Phi$ around $(\bar{x},\bar{y})$ given by

$\sup\big{\\{}\|x^{*}\|\;\big{|}\;x^{*}\in
D^{*}\Phi(\bar{x},\bar{y})(y^{*})\big{\\}}\leq\ell\|y^{*}\|\;\mbox{ for all
}\;y^{*}\in\mathbb{R}^{m},$ (4.9)

which is known as the coderivative/Mordukhovich criterion; see (m93, , Theorem
5.7) and (rw, , Theorem 9.40). Similarly to the case of locally Lipschitzian
multifunctions, we can equivalently reformulate (4.9) via the neighborhood
estimates in (4.7) but valid now for all $x\in\mathbb{B}_{\eta}(\bar{x})$,
$y\in\Phi(x)\cap\mathbb{B}_{\eta}(\bar{y})$, and $y^{*}\in\mathbb{R}^{m}$
without the uniform boundedness assumption on $\Phi$ around $\bar{x}$.

The situation with random multifunctions is essentially more involved in
comparison with the deterministic case, being rather similar in some aspects
while significantly different in the others; this can be precisely seen from
the results given below. Let us start with a coderivative characterization of
integrably locally Lipschitzian multifunctions (4.2) defined on general
measure spaces. The following theorem not only extends the coderivative
conditions in (4.7) to random mappings, but also provides a new result in the
deterministic case by dropping the uniform boundedness assumption. For
brevity, we prove a regular coderivative characterization similar to the
second estimate in (4.7). The one in the form of the first estimate therein
can be derived similarly to the proof of Proposition 4 given below.

######  Theorem 4.1 (coderivative characte rization of integrably locally
Lipschitzian multifunctions)

Let $\Phi\colon T\times{\mathbb{R}^{n}}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ be a set-valued normal integrand
defined on a complete finite measure space $(T,\mathcal{A},\mu)$, and let
$\bar{x}\in\mbox{\rm dom}\,\mathrm{E}_{\Phi}$. Then $\Phi$ is integrably
locally Lipschitzian around $\bar{x}$ if and only if there exist $\eta>0$,
$\ell\in\textnormal{L}^{1}(T,\mathbb{R}_{+})$, and $\widehat{T}\in\mathcal{A}$
with $\mu(T\backslash\widehat{T})=0$ such that

$\sup\big{\\{}\|x^{*}\|\;\big{|}\;x^{*}\in\widehat{D}^{*}\Phi_{t}(x,y)(y^{*})\big{\\}}\leq\ell(t)\|y^{*}\|\;\mbox{
for all }\;y\in\Phi_{t}(x)\;\mbox{ and }\;t\in\widehat{T}$ (4.10)

whenever $x\in\mathbb{B}_{\eta}(\bar{x})$ and $y^{*}\in\mathbb{R}^{m}$.

Proof. First we show that the fulfillment of condition (4.10) for all
$x\in\mathbb{B}_{\eta}(\bar{x})$ and $y^{*}\in\mathbb{R}^{m}$ yields the
integrable local Lipschitz property of $\Phi$ around $\bar{x}$. Take $\eta>0$,
$\ell\in\textnormal{L}^{1}(T,\mathbb{R}_{+})$, and $\widehat{T}\in\mathcal{A}$
with $\mu(T\backslash\widehat{T})$ for which (4.10) holds, and then fix
$t\in\widehat{T}$. Picking $x_{0},x_{1}\in\mathbb{B}_{\eta/2}(\bar{x})$ and
$y\in\Phi_{t}(x_{0})$, we claim that

$y\in\Phi_{t}(x_{1})+\ell(t)\|x_{0}-x_{1}\|\mathbb{B}.$ (4.11)

To proceed, define $x_{\lambda}:=(1-\lambda)x_{0}+\lambda x_{1}$ with
$\lambda\in[0,1]$ and denote

$\bar{\lambda}:=\sup\big{\\{}\lambda>0\;\big{|}\;y\in\Phi_{t}(x_{\lambda})+\ell(t)\lambda\|x_{0}-x_{1}\|\mathbb{B}\big{\\}}.$
(4.12)

Since $t$ is fixed, we can consider $\Phi_{t}(x)$ as a deterministic
multifunction and thus deduce from the aforementioned version of (4.7) as a
characterization of the Lipschitz-like property of deterministic
multifunctions by (m06, , Theorem 4.7) that $\Phi_{t}$ is Lipschitz-like
around $(x_{0},y)$. This gives us $\eta_{x_{0},y}>0$ such that

$y\in\Phi_{t}(x_{0})\cap\mathbb{B}_{\eta_{x_{0},y}}(y)\subset\Phi_{t}(u)+\ell(t)\|x_{0}-u\|\mathbb{B}\;\text{
for all }\;u\in\mathbb{B}_{\eta_{x_{0},y}}(x_{0}).$

Using the above inclusion for $u=x_{\lambda}$ with a sufficiently small number
$\lambda>0$ shows that $\bar{\lambda}>0$. Let us show that the supremum in
(4.12) is attained, i.e., we have

$\exists y_{\bar{\lambda}}\in\Phi_{t}(x_{\bar{\lambda}}),\;\exists
v_{\bar{\lambda}}\in\mathbb{B}\;\text{ such that
}\;y=y_{\bar{\lambda}}+\ell(t)\bar{\lambda}\|x_{0}-x_{1}\|v_{\bar{\lambda}}.$
(4.13)

Indeed, consider a sequence $\lambda_{k}\uparrow\bar{\lambda}$ together with
the points $y_{\lambda_{k}}\in\Phi_{t}(x_{\lambda_{k}})$ and
$v_{k}\in\mathbb{B}$ satisfying

$y=y_{\lambda_{k}}+\ell(t){\lambda_{k}}\|x_{0}-x_{1}\|v_{k},\quad
k\in\mathbb{N}.$ (4.14)

It follows that $x_{\lambda_{k}}\to x_{\bar{\lambda}}$ as $k\to\infty$, and
that $v_{k}\to v_{\bar{\lambda}}\in\mathbb{B}$ along a subsequence. Thus we
deduce from (4.14) that a subsequence of $\\{y_{\lambda_{k}}\\}$ converges to
some $y_{\bar{\lambda}}$, which yields (4.13) by the closedness of
$\operatorname{gph}\Phi_{t}$.

Applying now (4.10) and the characterization of (m06, , Theorem 4.7) tells us
that $\Phi_{t}$ is Lipschitz-like around
$(x_{\bar{\lambda}},y_{\bar{\lambda}})$, i.e., there exists
$\eta_{\bar{\lambda}}>0$ ensuring

$y_{\bar{\lambda}}\in\Phi_{t}(x_{\bar{\lambda}})\cap\mathbb{B}_{\eta_{\bar{\lambda}}}(y_{\bar{\lambda}})\subset\Phi_{t}(w)+\ell(t)\|w-x_{\bar{\lambda}}\|\mathbb{B}\;\mbox{
for all }\;w\in\mathbb{B}_{\eta_{\bar{\lambda}}}(x_{\bar{\lambda}}).$

The latter allows us to find $\lambda^{\prime}>\bar{\lambda}$ for which
$x_{\lambda^{\prime}}\in\mathbb{B}(x_{\bar{\lambda}})$,
$y_{\lambda^{\prime}}\in\Phi_{t}(x_{\lambda^{\prime}})$, and
$v_{\lambda^{\prime}}\in\mathbb{B}$ are such that

$y_{\bar{\lambda}}\in
y_{\lambda^{\prime}}+\ell(t)\|x_{\lambda^{\prime}}-x_{\bar{\lambda}}\|v_{\lambda^{\prime}}=y_{\lambda^{\prime}}+\ell(t)(\lambda^{\prime}-\bar{\lambda})\|x_{1}-x_{0}\|v_{\lambda^{\prime}}.$
(4.15)

It readily follows from (4.13) and (4.15) that

$\displaystyle y$
$\displaystyle=y_{\bar{\lambda}}+\ell(t)\bar{\lambda}\|x_{0}-x_{1}\|v_{\bar{\lambda}}$
$\displaystyle=y_{\lambda^{\prime}}+\ell(t)(\lambda^{\prime}-\bar{\lambda})\|x_{1}-x_{0}\|v_{\lambda^{\prime}}+\ell(t)\bar{\lambda}\|x_{0}-x_{1}\|v_{\bar{\lambda}}$
$\displaystyle=y_{\lambda^{\prime}}+\ell(t)\lambda^{\prime}\|x_{1}-x_{0}\|\left(\frac{\lambda^{\prime}-\bar{\lambda}}{\lambda^{\prime}}v_{\lambda^{\prime}}+\frac{\bar{\lambda}}{\lambda^{\prime}}v_{\bar{\lambda}}\right)$
$\displaystyle\in\Phi_{t}(x_{\lambda^{\prime}})+\ell(t)\lambda^{\prime}\|x_{0}-x_{1}\|\mathbb{B},$

which contradicts the maximality of $\bar{\lambda}$ as chosen in (4.12). This
gives us (4.11) and verifies therefore that $\Phi$ is integrably locally
Lipschitzian around $\bar{x}$.

Conversely, suppose that $\Phi$ is integrably locally Lipschitzian around
$\bar{x}$ and thus find $\eta>0$,
$\ell\in\textnormal{L}^{1}(T,\mathbb{R}_{+})$, and $\widehat{T}\in\mathcal{A}$
with $\mu(T\backslash\widehat{T})$ such that (4.2) is satisfied. Taking
$x\in\mathbb{B}_{\eta/2}(\bar{x})$, $t\in\widehat{T}$, and $y\in\Phi_{t}(x)$,
we deduce from (4.2) and the characterization of (m06, , Theorem 4.7) that

$\sup\big{\\{}\|x^{\ast}\|\;\big{|}\;x^{\ast}\in\widehat{D}^{\ast}\Phi_{t}(x,y)(y^{\ast})\big{\\}}\leq\ell(t)\|y^{\ast}\|\;\mbox{
for all }\;y^{\ast}\in\mathbb{R}^{m},$

which concludes the proof of the theorem. $\hfill\square$

Now we are ready to introduce two graphically localized Lipschitzian
properties of multifunctions defined on finite measure spaces. They both may
be considered as extensions to random multifunctions of the Lipschitz-like
property (4.8) and its coderivative characterizations for deterministic
multifunctions while being generally different from each other in the
stochastic case.

To proceed, we associate with a given set-valued integrand $\Phi\colon
T\times{\mathbb{R}^{n}}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ the multifunction ${\cal
S}\colon{\mathbb{R}^{n}}\times\mathbb{R}^{m}\to\textnormal{L}^{1}(T,\mathbb{R}^{m})$
defined by

$\mathcal{S}_{\Phi}(x,y):=\Big{\\{}\mathpzc{y}\in\textnormal{L}^{1}(T,\mathbb{R}^{m})\;\Big{|}\;\int_{T}\mathpzc{y}(t)d\mu=y\;\text{
and }\;\mathpzc{y}(t)\in\Phi_{t}(x)\;\text{ for a.e. }\;t\in T\Big{\\}}.$
(4.16)

######  Definition 3 (graphically localized random Lipschitzian
multifunctions)

Let $\Phi\colon T\times{\mathbb{R}^{n}}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ be a set-valued normal integrand
on a complete finite measure space $(T,\mathcal{A},\mu)$, let
$(\bar{x},\bar{y})\in\operatorname{gph}\mathrm{E}_{\Phi}$ for $E_{\Phi}$ taken
from (1.2), and let $\bar{\mathpzc{y}}\in\mathcal{S}_{\Phi}(\bar{x},\bar{y})$
for ${\cal S}_{\Phi}$ taken from (4.16). We say that:

(i) $\Phi$ is integrably Lipschitz-like around $(\bar{x},\bar{\mathpzc{y}})$
if there exist $\ell,\in\textnormal{L}^{1}(T,\mathbb{R}_{+})$, strictly
positive measurable functions $\eta$ and $\gamma$, and a measurable set
$\widehat{T}\in\mathcal{A}$ with $\mu(T\backslash\widehat{T})=0$ such that

$\Phi_{t}(x)\cap\mathbb{B}_{\gamma(t)}\big{(}\mathpzc{y}(t)\big{)}\subset\Phi_{t}(x^{\prime})+\ell(t)\|x-x^{\prime}\|\mathbb{B}\;\text{
for all }\;t\in\widehat{T}\;\text{ and
}\;x,x^{\prime}\in\mathbb{B}_{\eta(t)}(\bar{x}).$ (4.17)

(ii) $\Phi$ is integrably quasi-Lipschitzian around
$(\bar{x},\bar{\mathpzc{y}})$ if there exist $\eta>0$ and
$\ell\in\textnormal{L}^{1}(T,\mathbb{R}_{+})$ such that

$\sup\big{\\{}\|x^{\ast}\|\;\big{|}\;x^{\ast}\in{D}^{\ast}\Phi_{t}\big{(}\mathpzc{x}(t),\mathpzc{y}(t)\big{)}\big{(}\mathpzc{y}^{\ast}(t)\big{)}\big{\\}}\leq\ell(t)\|\mathpzc{y}^{\ast}(t)\|\text{
for a.e. }\;t\in T$ (4.18)

whenever $\mathpzc{x}\in\mathbb{B}_{\eta}(\bar{x})$,
$\mathpzc{y}\in\mathbb{B}_{\eta}(\bar{\mathpzc{y}})\cap\Phi(\mathpzc{x})$, and
$\mathpzc{y}^{\ast}\in\textnormal{L}^{\infty}(T,\mathbb{R}^{m})$ with

$\mathbb{B}_{\eta}(\bar{\mathpzc{y}})\cap\Phi(\mathpzc{x}):=\big{\\{}\mathpzc{y}\in\textnormal{L}^{1}(T,\mathbb{R}^{m})\;\big{|}\;\mathpzc{y}\in\mathbb{B}_{\eta}(\bar{\mathpzc{y}})\;\text{
and }\;\mathpzc{y}(t)\in\Phi_{t}\big{(}\mathpzc{x}(t)\big{)}\;\text{
a.e.}\big{\\}}.$

First we show that the integrable quasi-Lipschitzian property can be
equivalently reformulated in terms of the regular coderivative (2.4) replacing
the basic one in (4.18).

######  Proposition 4 (equivalent description of the integrable quasi-
Lipschitzian property)

Let $\Phi\colon T\times{\mathbb{R}^{n}}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ be a set-valued normal integrand
with $(\bar{x},\bar{y})\in\operatorname{gph}\mathrm{E}_{\Phi}$ and
$\bar{\mathpzc{y}}\in\mathcal{S}_{\Phi}(\bar{x},\bar{y})$. Then $\Phi$ is
integrably quasi-Lipschitzian around $(\bar{x},\bar{\mathpzc{y}})$ if and only
if there exist $\eta>0$ and $\ell\in\textnormal{L}^{1}(T,\mathbb{R}_{+})$ such
that

$\sup\big{\\{}\|x^{\ast}\|\;\big{|}\;x^{\ast}\in\widehat{D}^{\ast}\Phi_{t}\big{(}\mathpzc{x}(t),\mathpzc{y}(t)\big{)}\big{(}\mathpzc{y}^{\ast}(t)\big{)}\big{\\}}\leq\ell(t)\|\mathpzc{y}^{\ast}(t)\|\;\text{
for a.e. }\;t\in T$ (4.19)

whenever $\mathpzc{x}\in\mathbb{B}_{\eta}(\bar{x})$,
$\mathpzc{y}\in\mathbb{B}_{\eta}(\bar{\mathpzc{y}})\cap\Phi(\mathpzc{x})$, and
$\mathpzc{y}^{\ast}\in\textnormal{L}^{\infty}(T,\mathbb{R}^{m})$.

Proof. We obviously have that (4.2) yields (4.19). To verify the opposite
implication, suppose without loss of generality that $\mu(T)=1$ and take
$\eta>0$ and $\ell\in\textnormal{L}^{1}(T,\mathbb{R}_{+})$ from (4.19). Fix
further $\gamma\in(0,\eta)$ and pick
$\mathpzc{x}\in\mathbb{B}_{\gamma}(\bar{x})$,
$\mathpzc{y}\in\mathbb{B}_{\gamma}(\bar{\mathpzc{y}})\cap\Phi(\mathpzc{x})$,
and $\mathpzc{y}^{\ast}\in\textnormal{L}^{\infty}(T,\mathbb{R}^{m})$.
Proposition 1 tells us that the set-valued mapping
$F(t):={D}^{\ast}\Phi_{t}(\mathpzc{x}(t),\mathpzc{y}(t))(\mathpzc{y}^{\ast}(t))$
is measurable with closed values on $T$. By the Castaing representation of the
measurable multifunction $F$ (see, e.g., (rw, , Theorem 14.5)) we find a
sequence of measurable selections $\mathpzc{x}_{k}^{\ast}(t)\in F(t)$ for a.e.
$t\in\widehat{T}:=\mbox{\rm dom}\,F$ such that

$\sup\big{\\{}\|x_{k}^{\ast}(t)\|\;\big{|}\;k\in\mathbb{N}\big{\\}}=\sup\big{\\{}\|x^{\ast}\|\;\big{|}\;x^{\ast}\in{D}^{\ast}\Phi_{t}\big{(}\mathpzc{x}(t),\mathpzc{y}(t)\big{)}\big{(}\mathpzc{y}^{\ast}(t)\big{)}\big{\\}}\;\text{
for a.e. }\;t\in\widehat{T}.$

Fixing any $\varepsilon\in(0,(\eta-\gamma)/2)$ and $k\in\mathbb{N}$, define
the multifunction $F^{k}_{\varepsilon}\colon\widehat{T}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{2(n+m)}$ by

$(u,u^{\ast},v,v^{\ast})\in
F^{k}_{\varepsilon}(t)\Longleftrightarrow\left\\{\begin{array}[]{ll}u^{\ast}\in\widehat{D}^{\ast}\Phi_{t}(u,v)(v^{\ast}),\;u\in\mathbb{B}_{\gamma}\big{(}\mathpzc{x}(t)\big{)},\;v\in\mathbb{B}_{\gamma}\big{(}\mathpzc{y}(t)\big{)},\\\
\|u^{\ast}-\mathpzc{x}_{k}^{\ast}(t)\|\leq\varepsilon,\;\|v^{\ast}-\bar{\mathpzc{y}}^{\ast}(t)\|\leq\varepsilon/(1+\ell(t)).\end{array}\right.$

It follows from Proposition 1 and the measurability of the mappings involved
that the multifunctions $F^{k}_{\varepsilon}$ are graph measurable. Applying
the coderivative representation (2.7) to $F_{\varepsilon}^{k}(\cdot)$ ensures
that the sets $F_{\varepsilon}^{k}(t)$ are nonempty for a.e.
$t\in\widehat{T}$. Then Proposition 2 gives us a measurable selection
$(\mathpzc{u}(t),\mathpzc{u}^{\ast}(t),\mathpzc{v}(t),\mathpzc{v}^{\ast}(t))\in
F_{\varepsilon}^{k}(t)$ for a.e. $t\in\widehat{T}$ and $\varepsilon,k$ fixed
above. Thus $\mathpzc{u}\in\mathbb{B}_{\eta}(\bar{x})$,
$\mathpzc{v}\in\mathbb{B}_{\eta}(\bar{\mathpzc{y}})\cap\Phi(\mathpzc{u})$, and
$\mathpzc{v}^{\ast}\in\textnormal{L}^{\infty}(T,\mathbb{R}^{m})$. Employing
the regular coderivative estimate (4.19) ensures the relationships

$\displaystyle\|\mathpzc{x}_{k}^{\ast}(t)\|$
$\displaystyle\leq\|\mathpzc{u}^{\ast}(t)\|+\varepsilon\leq\ell(t)\|\mathpzc{v}^{\ast}(t)\|+\varepsilon\leq\ell(t)\|\bar{\mathpzc{y}}^{\ast}(t)\|+\ell(t)\|\mathpzc{v}^{\ast}(t)-\bar{\mathpzc{y}}^{\ast}(t)\|+\varepsilon$
$\displaystyle\leq\ell(t)\|\bar{\mathpzc{y}}^{\ast}(t)\|+2\varepsilon\;\mbox{
a.e. }\;t\in\widehat{T}.$

Since $\varepsilon\in(0,(\eta-\gamma)/2)$ and $k\in\mathbb{N}$ were chosen
arbitrarily, we arrive at the integrable quasi-Lipschitzian property (4.18)
and thus complete the proof of the proposition. $\hfill\square$

Next we establish closed relationships (actually the equivalence) between the
graphically localized Lipschitzian properties of random multifunctions from
Definition 3 in the case of purely atomic spaces with countably many atoms.
The following theorem can be treated as a stochastic extension of the
coderivative characterization of the Lipschitz-like property for deterministic
multifunctions.

######  Theorem 4.2 (relationships between graphically localized integrably
Lipschitzian properties)

Let $(T,\mu,\mathcal{A})$ be a purely atomic measure space consisting of
countable disjoint family of atoms $(T_{k})_{k\in\mathbb{N}}$. Consider a set-
valued normal integrand $\Phi\colon T\times{\mathbb{R}^{n}}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ with
$(\bar{x},\bar{y})\in\operatorname{gph}\mathrm{E}_{\Phi}$ and
$\bar{\mathpzc{y}}\in\mathcal{S}_{\Phi}(\bar{x},\bar{y})$. If $\Phi$ is
integrably quasi-Lipschitzian around $(\bar{x},\bar{\mathpzc{y}})$, then there
exist $\gamma>0$ and $\ell\in\textnormal{L}^{1}(T,\mathbb{R}_{+})$ such that
for all $k\in\mathbb{N}$ we have the integrable Lipschitz-like property in the
form

$\Phi_{t}(x)\cap\mathbb{B}_{\frac{\gamma}{\mu(T_{k})}}\big{(}\bar{y}(t)\big{)}\subset\Phi_{t}(x^{\prime})+\ell(t)\|x-x^{\prime}\|\mathbb{B}\;\text{
for all
}\;x,x^{\prime}\in\mathbb{B}_{\frac{\gamma}{(\ell(t)+1)}}(\bar{x})\;\text{ and
}\;t\in T_{k}.$ (4.20)

Conversely, the fulfillment of the integrable Lipschitz-like property (4.20)
with some $\ell\in\textnormal{L}^{\infty}(T,\mathbb{R}_{+})$ implies that
$\Phi$ is integrably quasi-Lipschitzian around $(\bar{x},\bar{y})$.

Proof. Suppose that $\Phi$ is integrably quasi-Lipschitzian around
$(\bar{x},\bar{\mathpzc{y}})$ with the given data $\eta>0$ and
$\ell\in\textnormal{L}^{1}(T,\mathbb{R}_{+})$ for which (4.18) holds. Since
the measure is purely atomic, each measurable function or multifunction must
be constant a.e. on each atom. Then we can assume that all the involved
mappings are constants over the atoms, i.e., $\Phi_{t}=\Phi_{k}$,
$\bar{\mathpzc{y}}(t)=\bar{y}_{k}$, and $\ell(t)=\ell_{k}$ on $T_{k}$.
Furthermore, observe that the coderivative condition (4.19) implies that for
every $k\in\mathbb{N}$ we have

$\displaystyle\sup\big{\\{}\|x^{\ast}\|\;\big{|}\;x^{\ast}\in\widehat{D}^{\ast}\Phi_{k}(x,y)(y^{\ast})\big{\\}}\leq\ell_{k}\|y^{\ast}\|$
(4.21)

whenever $x\in\mathbb{B}_{\eta}(\bar{x}_{k})$,
$y\in\mathbb{B}_{\frac{\eta}{\mu(T_{k})}}(\bar{y}_{k})\cap\Phi_{k}(x)$, and
$y^{\ast}\in\mathbb{R}^{m}$. Denote $\gamma:=\frac{\eta}{3(\mu(T)+1)}$ and fix
$k\in\mathbb{N}$. Then consider
$x_{0},x_{1}\in\mathbb{B}_{\frac{\gamma}{(\ell_{k}+1)}}(\bar{x})$ and
$y_{k}\in\Phi_{k}(x_{0})\cap\mathbb{B}_{\frac{\gamma}{\mu(T_{k})}}(\bar{y}_{k})$.
We claim that

$y_{k}\in\Phi_{k}(x_{1})+\ell_{k}\|x_{0}-x_{1}\|\mathbb{B}.$ (4.22)

Indeed, define $x_{\lambda}:=(1-\lambda)x_{0}+\lambda x_{1}$ for
$\lambda\in[0,1]$ and

$\bar{\lambda}:=\sup\big{\\{}\lambda>0\;\big{|}\;y_{k}\in\Phi_{k}(x_{\lambda})+\ell_{k}\lambda\|x_{0}-x_{1}\|\mathbb{B}\big{\\}}.$

Applying the coderivative criterion (4.21) for the deterministic
multifunctions $\Phi_{k}$, $k\in\mathbb{N}$, we have by (m06, , Theorem 4.7)
that $\Phi_{k}$ is Lipschitz-like around $(x_{0},y_{k})$. This gives us
$\eta_{k}>0$ such that

$y_{k}\in\Phi_{k}(x_{0})\cap\mathbb{B}_{\eta_{k}}(y_{k})\subset\Phi_{k}(u)+\ell_{k}\|x_{0}-u\|\mathbb{B}\;\text{
for all }\;u\in\mathbb{B}_{\eta_{k}}(x_{0}).$ (4.23)

Plugging there $u=x_{\lambda}$ for sufficiently small $\lambda>0$ ensures that
$\bar{\lambda}>0$. Arguing similarly to the proof of Theorem 4.1 verifies that
the supremum in the definition of $\bar{\lambda}$ is realized, i.e.,

$\exists y_{\bar{\lambda}}\in\Phi_{k}(x_{\bar{\lambda}}),\;\exists
v_{\bar{\lambda}}\in\mathbb{B}\;\text{ such that
}\;y_{k}=y_{\bar{\lambda}}+\ell_{k}\bar{\lambda}_{k}\|x_{0}-x_{1}\|v_{\bar{\lambda}}.$
(4.24)

Moreover, we have the following estimate of the distance between $\bar{y}_{k}$
and $y_{\bar{\lambda}}$ from (4.24):

$\|\bar{y}_{k}-y_{\bar{\lambda}}\|\leq\frac{2}{3\mu(T_{k})}\eta.$

Using (4.21) and applying again (m06, , Theorem 4.7) tell us that each
multifunction $\Phi_{k}$ is Lipschitz-like around
$(x_{\bar{\lambda}},y_{\bar{\lambda}})$. Thus there exists
$\eta_{\bar{\lambda}}>0$ ensuring the inclusions

$y_{\bar{\lambda}}\in\Phi_{k}(x_{\bar{\lambda}})\cap\mathbb{B}_{\eta_{\bar{\lambda}}}(y_{\bar{\lambda}})\subset\Phi_{k}(w)+\ell_{k}\|w-x_{\bar{\lambda}}\|\mathbb{B}\;\text{
for all }\;w\in\mathbb{B}_{\eta_{\bar{\lambda}}}(x_{\bar{\lambda}}).$ (4.25)

The latter yields the existence of $\lambda^{\prime}>\bar{\lambda}$ such that
$x_{\lambda^{\prime}}\in\mathbb{B}(x_{\bar{\lambda}})$,
$y_{\lambda^{\prime}}\in\Phi_{k}(x_{\lambda^{\prime}})$, and
$v_{\lambda^{\prime}}\in\mathbb{B}$ for which

$y_{\bar{\lambda}}\in
y_{\lambda^{\prime}}+\ell_{k}\|x_{\lambda^{\prime}}-x_{\bar{\lambda}}\|v_{\lambda^{\prime}}=y_{\lambda^{\prime}}+\ell_{k}(\lambda^{\prime}-\bar{\lambda})\|x_{1}-x_{0}\|v_{\lambda^{\prime}}.$

Using then (4.24) and (4.25) implies that

$\displaystyle y$
$\displaystyle=y_{\bar{\lambda}}+\ell_{k}\bar{\lambda}\|x_{0}-x_{1}\|v_{\bar{\lambda}}=y_{\lambda^{\prime}}+\ell_{k}(\lambda^{\prime}-\bar{\lambda})\|x_{1}-x_{0}\|v_{\lambda^{\prime}}+\ell_{k}\bar{\lambda}\|x_{0}-x_{1}\|v_{\bar{\lambda}}$
$\displaystyle=y_{\lambda^{\prime}}+\ell_{k}\lambda^{\prime}\|x_{1}-x_{0}\|\left(\frac{\lambda^{\prime}-\bar{\lambda}}{\lambda^{\prime}}v_{\lambda^{\prime}}+\frac{\bar{\lambda}}{\lambda^{\prime}}v_{\bar{\lambda}}\right)\in\Phi_{k}(x_{\lambda^{\prime}})+\ell_{k}\lambda^{\prime}\|x_{0}-x_{1}\|\mathbb{B},$

which contradicts the above choice of $\bar{\lambda}$. This tells us that
(4.22) holds, and thus $\Phi$ satisfies the integrable Lipschitz-like property
(4.20).

Conversely, assume that $\Phi$ satisfies (4.20) with some
$\ell\in\textnormal{L}^{\infty}(T,\mathbb{R}_{+})$. A close look at the proof
of (m06, , Theorem 1.43) tells us that (4.20) implies that

$\sup\big{\\{}\|x^{\ast}\|\;\big{|}\;x^{\ast}\in\widehat{D}^{\ast}\Phi_{t}(x,y)(y^{\ast})\big{\\}}\leq\ell_{k}\|y^{\ast}\|\;\text{
on }\;T_{k}$ (4.26)

whenever $x\in\mathbb{B}_{\frac{\gamma}{2(\ell_{k}+1)}}(\bar{x})$,
$y\in\mathbb{B}_{\frac{\gamma}{2\mu(T_{k})}}(\bar{\mathpzc{y}}(t))\cap\Phi_{t}(x)$,
and $y^{\ast}\in\mathbb{R}^{m}$. Picking now any number
$0<\eta<\frac{\gamma}{2(\|\ell\|_{\infty}+1)}$ together with measurable
functions $\mathpzc{x}\in\mathbb{B}_{\eta}(\bar{x})$,
$\mathpzc{y}\in\mathbb{B}_{\eta}(\bar{\mathpzc{y}})\cap\Phi(\mathpzc{x})$, and
$\mathpzc{y}^{\ast}\in\textnormal{L}^{\infty}(T,\mathbb{R}^{m})$ yields

$\|\mathpzc{y}(t)-\bar{\mathpzc{y}}(t)\|\leq\frac{\gamma}{2\mu(T_{k})}\;\mbox{
for a.e. }\;t\in T_{k}.$

Then the application of (4.26) verifies the integrable quasi-Lipschitzian
property of $\Phi$ around $(\bar{x},\bar{\mathpzc{y}})$. $\hfill\square$

Now we discover a remarkable phenomenon concerning Lipschitzian properties of
random multifunctions, which does not have any analogs in the deterministic
framework. Namely, it is revealed that in spaces with nonatomic measures
integrable quasi-Lipschitzian and local Lipschitzian properties of random
multifunctions agree. Furthermore, the measure nonatomicity is also necessary
for the fulfillment of this phenomenon as far as arbitrary multifunctions with
these properties are addressed.

######  Theorem 4.3 (integrably Lipschitzian multifunctions on nonatomic
spaces)

Let $(T,\mathcal{A},\mu)$ be a complete finite measure spaces, and let
$\Phi\colon T\times{\mathbb{R}^{n}}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ be a set-valued normal integrand.
Assume that the measure $\mu$ in nonatomic. Then given any
$(\bar{x},\bar{y})\in\operatorname{gph}\mathrm{E}_{\Phi}$ and
$\bar{\mathpzc{y}}\in\mathcal{S}_{\Phi}(\bar{x},\bar{y})$, we have that $\Phi$
is integrably quasi-Lipschitzian around $(\bar{x},\bar{\mathpzc{y}})$ if and
only if it is integrably locally Lipschitzian around $\bar{x}$. Conversely,
the presence of at least one atom on $T$ yields the existence of a
multifunction $\Phi\colon T\times{\mathbb{R}^{n}}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$, which is integrably quasi-
Lipschitzian around $(\bar{x},\bar{\mathpzc{y}})$ but not integrably locally
Lipschitzian around $\bar{x}$.

Proof. It follows from the equivalent descriptions of the integrably locally
Lipschitzian and quasi-Lipschitzian properties in Theorem 4.1 and Proposition
4, respectively, that the former property implies the latter one without the
nonatomicity assumption. Assume now that $\Phi$ is integrably quasi-
Lipschitzian around the point in question. Arguing by contraposition, suppose
that $\Phi$ is integrably locally Lipschitzian around $\bar{x}$, i.e., the
coderivative condition (4.10) in Theorem 4.1 fails. This gives us a set
$A\in\mathcal{A}$ with $\mu(A)>0$ such that for all $t\in A$ there exist
points $x_{t}\in\mathbb{B}_{\eta}(\bar{x})$, $y_{t}^{\ast}\in\mathbb{R}^{m}$,
$y_{t}\in\Phi_{t}(x_{t})$, and
$x_{t}^{\ast}\in\widehat{D}^{\ast}\Phi_{t}(x_{t},y_{t})(y_{t}^{\ast})$ with
$\|x^{\ast}_{t}\|>\ell(t)\|y_{t}^{\ast}\|$. Now define the multifunction
$F\colon A\;{\lower 1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ by

$\displaystyle\operatorname{gph}F:=\left\\{(t,x,y,x^{\ast},y^{\ast})\in
T\times\mathbb{R}^{2(n+m)}\;\Bigg{|}\;\begin{array}[]{ll}x\in\mathbb{B}_{\eta}(\bar{x}),\;(t,x,y)\in\operatorname{gph}\Phi\\\
(x^{\ast},-y^{\ast})\in\widehat{N}_{\operatorname{gph}\Phi_{t}}(x,y),\\\
\text{and }\;\|x^{\ast}\|>\ell(t)\|y^{\ast}\|\end{array}\right\\}.$

Observe that $F$ is graph measurable with nonempty (not necessary closed)
values. The measurable selection theorem presented in Proposition 2 gives us
measurable functions $\mathpzc{x},\mathpzc{x}^{\ast}\colon
T\to{\mathbb{R}^{n}}$ and $\mathpzc{y},\mathpzc{y}^{\ast}\colon
T\to\mathbb{R}^{m}$ such that for almost all $t\in A$ we get

$\mathpzc{x}(t)\in\mathbb{B}_{\eta}(\bar{x}),\;\mathpzc{x}^{\ast}(t)\in\widehat{D}^{\ast}\Phi_{t}\big{(}\mathpzc{x}(t),\mathpzc{y}(t)\big{)}\big{(}\mathpzc{y}^{\ast}(t)\big{)}\;\mbox{
and }\;\|\mathpzc{x}^{\ast}(t)\|>\ell(t)\|\mathpzc{y}^{\ast}(t)\|.$ (4.27)

The nonatomicity of $\mu$ allows us to find a measurable set $B\subset A$ with
$\mu(B)>0$ and $\int_{B}\|\mathpzc{y}(t)-\bar{\mathpzc{y}}(t)\|d\mu\leq\eta$.
Hence the measurable functions
$\mathpzc{v}:=\mathpzc{x}\operatorname{\mathds{1}}_{B}+\bar{x}\operatorname{\mathds{1}}_{B^{c}}$
and
$\mathpzc{w}:=\mathpzc{y}\operatorname{\mathds{1}}_{B}+\bar{\mathpzc{y}}\operatorname{\mathds{1}}_{B^{c}}$
belong to the balls $\mathbb{B}_{\eta}(\bar{x})$ and
$\mathbb{B}_{\eta}(\bar{\mathpzc{y}})$, respectively. Employing now (4.27)
with taking into account Proposition 4, we arrive at a contradiction with
(4.18) and thus verify the assertion of the theorem in nonatomic spaces.

To justify the converse assertion, suppose that $(T,\mathcal{A},\mu)$ contains
at least one atom, say $T_{0}\in\mathcal{A}$. Taking into account the
representation of the measure space as the union of purely atomic and
nonatomic parts as discussed in Section 3, we assume without loss of
generality that the space $T\backslash T_{0}$ is nonatomic; see also the proof
of Proposition 5 for more details in similar arguments. Consider two arbitrary
deterministic multifunctions $F_{1},F_{2}\colon{\mathbb{R}^{n}}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ such that $F_{1}$ is locally
Lipschitzian around some $\bar{x}$, while $F_{2}$ is Lipschitz-like around
$(\bar{x},\bar{y})$ with some $\bar{y}\in F_{2}(\bar{x})$ but not being
locally Lipschitzian around $\bar{x}$. Specifically $F_{2}$ can be constructed
as $F_{2}(x)=|x|$ on $\mathbb{R}$ with the additional value $F(0)=1$, while
$(\bar{x},\bar{y})=(0,0)$. Define now the random multifunction $\Phi\colon
T\times{\mathbb{R}^{n}}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ by

$\displaystyle\Phi_{t}(x):=\left\\{\begin{array}[]{ll}F_{1}(x)&\text{ if
}\;t\in T\backslash T_{0},\\\ F_{2}(x)&\text{ if }\;t\in T_{0}\\\
\end{array}\right.$ (4.30)

and construct a measurable selection $\bar{\mathpzc{y}}(t)\in\Phi_{t}(x)$ by
$\bar{\mathpzc{y}}(t):=y_{1}\operatorname{\mathds{1}}_{T\backslash
T_{0}}+\bar{y}_{1}\operatorname{\mathds{1}}_{T_{0}}$ with some $y_{1}\in
F_{1}(x)$. It follows from Theorem 4.1 and from the first part of this theorem
applied on the nonatomic part $T\backslash T_{0}$ that the multifunction
$\Phi$ from (4.30) is integrably quasi-Lipschitzian around
$(\bar{x},\bar{\mathpzc{y}})$ when $t\in T\backslash T_{0}$. This property of
$\Phi$ on the purely atomic part $T_{0}$ follows from Theorem 4.2 since
$F_{2}$ is a deterministic Lipschitz-like multifunction. However, the random
multifunction $\Phi$ cannot be integrably locally Lipschitzian around
$\bar{x}$ since $F_{2}$ in (4.30) was chosen to be merely Lipschitz-like
around the point in question. This therefore completes the proof of the
theorem. $\hfill\square$

It is easy to deduce from Theorem 4.3 that the integrably Lipschitz-like and
quasi-Lipschitzian properties are different in any nonatomic space. Indeed,
Theorem 4.3 tells us that in such spaces the latter property is equivalent to
its locally Lipschitzian counterpart defined by inclusion (4.2), which may
clearly be different from the integrable Lipschitz-like property (4.2)
whenever the sets $\Phi_{t}(x)$ are unbounded. The following simple example
demonstrates the difference between these two Lipschitzian properties of
random set-valued mappings even in the case of bounded multifunctions.

###### Example 2

$($quasi-Lipschitzian versus Lipschitz-like random multifunctions$)$ Consider
any nonatomic probability space $(T,\mathcal{A},\mu)$ and the constant mapping
$\Phi\colon T\times\mathbb{R}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}_{+}$ defined by
$\Phi(t,x):=[0,\sqrt{|x|}+1]$. Then we have
$\mathrm{E}_{\Phi}(x)=[0,\sqrt{|x|}+1]$ with
$(\bar{x},\bar{y})=(0,0)\in\operatorname{gph}\mathrm{E}_{\Phi}$ and the
constant function $\bar{\mathpzc{y}}(t)=0$ belongs to
$\mathcal{S}_{\Phi}(\bar{x},\bar{y})$ from (4.16). It is easy to see that
$\mathrm{E}_{\Phi}(x)$ is Lipschitz-like around $(\bar{x},\bar{y})$. However,
$\Phi$ cannot be quasi-Lipschitzian around this point, because otherwise it
follows from Theorem 4.1 and 4.3 that $\Phi$ satisfies (4.2). The latter leads
us to a contradiction, since it would imply, in particular, that the function
$\varphi(x):=\sqrt{|x|}$ is locally Lipschitzian around $\bar{x}=0$.

Similarly to the deterministic case, the integrable Lipschitz-like property
brings us to conclusions about Lipschitz stability of feasible solution sets
in stochastic programming, while the integrable quasi-Lipschitzian property
plays a crucial role in deriving the most useful pointwise Leibniz-type rules
for coderivatives and second-order subdifferentials that are established in
the subsequent sections.

The final result of this section provides efficient conditions ensuring the
integral quasi-Lipschitzian property of set-valued normal integrands defined
on general measure spaces that are important, in particular, to check the
qualification conditions in the calculus rules derived below.

######  Proposition 5 (sufficient conditions for integrably quasi-Lipschitzian
multifunctions)

Let $(T,\mathcal{A},\mu)$ be a complete finite measure space, and let
$\Phi\colon T\times{\mathbb{R}^{n}}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ be a set-valued normal integrand.
Take $(\bar{x},\bar{y})\in\operatorname{gph}\mathrm{E}_{\Phi}$ and
$\bar{\mathpzc{y}}\in\mathcal{S}_{\Phi}(\bar{x},\bar{y})$ and suppose that
there exist $\varepsilon,\ell_{pa}>0$ as well as $\gamma_{pa}>0$ and
$\ell_{na}\in\textnormal{L}^{1}(T,\mathbb{R_{+}})$ such that the following
inclusions hold for all $x,x^{\prime}\in\mathbb{B}_{\varepsilon}(\bar{x})$:

$\begin{array}[]{ll}\Phi_{t}(x)\cap\mathbb{B}_{\frac{\gamma_{pa}}{\mu(T_{k})}}\big{(}\bar{y}(t)\big{)}\subset\Phi_{t}(x^{\prime})+\ell_{pa}\|x-x^{\prime}\|\mathbb{B}\;\text{
if }\;t\in T^{k}_{pa},\\\
\Phi_{t}(x)\subset\Phi_{t}(x^{\prime})+\ell_{na}(t)\|x-x^{\prime}\|\mathbb{B}\;\text{
if }\;t\in T_{na},\end{array}$ (4.31)

where $T_{pa}$ and $T_{na}$ form a disjoint decomposition of $T$ into purely
atomic and nonatomic parts with $\\{T_{pa}^{k}\\}_{k\in\mathbb{N}}$ being a
countable disjoint family of atoms. Then $\Phi$ is quasi-Lipschitzian around
$(\bar{x},\bar{\mathpzc{y}})$.

Proof. Let $\mu_{pa}(\cdot)$ and $\mu_{na}(\cdot)$ be the purely atomic and
nonatomic part of $\mu$, i.e., $\mu_{pa}(\cdot):=\mu(\cdot\cap T_{pa})$ and
$\mu_{na}(\cdot):=\mu(\cdot\cap T_{na})$, , respectively. Using the second
inclusion in (4.31), we apply Theorems 4.1 and 4.3 to the multifunction $\Phi$
on the nonatomic measure space $(T_{na},\mathcal{A},\mu_{na})$ and thus find
$\eta_{na}>0$ and $\ell_{na}\in\textnormal{L}^{1}(T,\mathbb{R}_{+})$ such that
(4.18) holds. On the other hand, the first inclusion in (4.31) allows us to
apply Theorem 4.2 to $\Phi$ on the purely atomic measure space
$(T_{pa},\mathcal{A},\mu_{pa})$ and hence get $\eta_{pa}>0$ and
$\ell_{pa}\in\textnormal{L}^{1}(T,\mathbb{R}_{+})$ such that (4.18) holds. Now
we claim that $\Phi$ satisfies (4.18) with
$\eta:=\min\\{\eta_{na},\eta_{pa}\\}$ and
$\ell(t):=\ell_{na}(t)\operatorname{\mathds{1}}_{T_{na}}(t)+\ell_{pa}(t)\operatorname{\mathds{1}}_{T_{pa}}(t)$
on the entire measure space $(T,\mathcal{A},\mu)$. Indeed, picking any
$\mathpzc{x}\in\mathbb{B}_{\eta}(\bar{x})$,
$\mathpzc{y}\in\mathbb{B}_{\eta}(\bar{\mathpzc{y}})\cap\Phi(\mathpzc{x})$, and
$\mathpzc{y}^{\ast}\in\textnormal{L}^{\infty}(T,\mathbb{R}^{m})$ leads us to
the estimate

$\int_{T}\|\mathpzc{y}(t)-\bar{\mathpzc{y}}(t)\|d\mu_{na}+\int_{T}\|\mathpzc{y}(t)-\bar{\mathpzc{y}}(t)\|d\mu_{pa}=\int_{T}\|\mathpzc{y}(t)-\bar{\mathpzc{y}}(t)\|d\mu\leq\eta,$

which yields (4.18) for the selected triple
$(\mathpzc{x},\mathpzc{y},\mathpzc{y}^{\ast})$ and thus completes the proof of
the proposition. $\hfill\square$

##  5 Coderivative Leibniz Rules for Expected-Integral Mappings

In this section we establish several calculus rules to evaluate coderivatives
of the expected-integral multifunctions
$\mathrm{E}_{\Phi}\colon{\mathbb{R}^{n}}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ given by

$\mathrm{E}_{\Phi}(x):=\int_{T}\Phi_{t}(x)d\mu\;\mbox{ for all
}\;x\in{\mathbb{R}^{n}},$ (5.1)

where $\Phi\colon T\times{\mathbb{R}^{n}}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ is a set-valued normal integrand
defined on a complete finite measure space $(T,\mathcal{A},\mu)$. Various
results, unified under the name of coderivative Leibniz rules, are obtained to
evaluate the regular and limiting coderivatives of (5.1) via the corresponding
coderivatives of the integrand multifunctions $\Phi_{t}$. The most efficient
and useful pointwise rules will be obtained to evaluate the limiting
coderivative of $\mathrm{E}_{\Phi}$ at the given point $(\bar{x},\bar{y})$ in
terms of the limiting coderivative of $\Phi_{t}$ under the integrable quasi-
Lipschitzian property of the integrand multifunctions around the corresponding
points.

Throughout this section we assume that at the given point of interest
$\bar{x}\in\mbox{\rm dom}\,\mathrm{E}_{\Phi}$ there exist a positive number
$\rho$ and a function $\kappa\in\textnormal{L}^{1}(T,\mathbb{R}_{+})$ such
that the following conditions are satisfied:

$\displaystyle\Phi_{t}(x)$ $\displaystyle\text{ is convex for all
}\;x\in\mathbb{B}_{\rho}(\bar{x})\;\text{ and a.e. }\;t\in T_{na},$ (5.2)
$\displaystyle\Phi_{t}(x)$ $\displaystyle\subset\kappa(t)\mathbb{B}\;\text{
for all }\;x\in\mathbb{B}_{\rho}(\bar{x})\;\text{ and a.e. }\;t\in T,$

where the second condition is known as integrable boundedness.

We start with deriving sequential Leibniz-type rules to estimate the regular
coderivative of $\mathrm{E}_{\Phi}$ via sequences of regular coderivatives of
$\Phi_{t}$ at points nearby. These results are of the same flavor as the
sequential subdifferential Leibniz rules for expected-integral functionals
(1.1) that are presented in Theorems 3.1 and 3.2 and are used in the proofs
below. Since the proofs of the following two propositions are similar to each
other, we prove only the second one.

######  Proposition 6 (sequential coderivative Leibniz rule, I)

Let $\Phi\colon T\times{\mathbb{R}^{n}}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ be a set-valued normal integrand
defined on a complete finite measure space $(T,\mathcal{A},\mu)$, let
$\bar{x}\in\mbox{\rm dom}\,\mathrm{E}_{\Phi}$ satisfy the conditions in (5.2),
and let $\bar{\mathpzc{y}}\in\mathcal{S}_{\Phi}(\bar{x},\bar{y})$ be taken
from (4.16). Then for any fixed $p,q\in(1,\infty)$ with $1/p+1/q=1$ there
exist sequences $\\{x_{k}\\}\subset{\mathbb{R}^{n}}$,
$\\{\mathpzc{x}_{k}\\}\subset\textnormal{L}^{p}({T},{\mathbb{R}^{n}})$,
$\\{\mathpzc{x}_{k}^{*}\\}\subset{\textnormal{L}}^{q}({T},{\mathbb{R}^{n}})$,
$\\{\mathpzc{y}_{k}\\}\subset\textnormal{L}^{1}(T,\mathbb{R}^{m})$, and
$\\{\mathpzc{y}_{k}^{\ast}\\}\subset\textnormal{L}^{\infty}(T,\mathbb{R}^{m})$
such that we have the assertions:

  1. (i)

$\mathpzc{x}_{k}^{*}(t)\in\widehat{D}^{\ast}\Phi_{t}\big{(}\mathpzc{x}_{k}(t),\mathpzc{y}_{k}(t)\big{)}\big{(}\mathpzc{y}_{k}^{\ast}(t)\big{)}$
for a.e. $t\in T$ and all $k\in\mathbb{N}$.

  2. (ii)

$\|\bar{x}-x_{k}\|\to 0$, $\|\bar{x}-\mathpzc{x}_{k}\|_{p}\to 0$, and
$\|\bar{\mathpzc{y}}-\mathpzc{y}_{k}\|_{1}\to 0$ as $k\to\infty$.

  3. (iii)

$\|\mathpzc{y}_{k}^{\ast}-\bar{\mathpzc{y}}^{\ast}\|_{\infty}\to 0$,
$\|\mathpzc{x}_{k}^{*}\|_{q}\|\mathpzc{x}_{k}-x_{k}\|_{p}\to 0$, and
$\displaystyle\int_{T}x_{k}^{*}(t)d\mu\to\bar{x}^{\ast}$ as $k\to\infty$.

######  Proposition 7 (sequential coderivative Leibniz rule, II)

Let
$\bar{x}^{\ast}\in\widehat{D}^{\ast}\mathrm{E}_{\Phi}(\bar{x},\bar{y})(\bar{y}^{\ast})$
in the setting of Proposition 7. Then there exist sequences
$\\{x_{k}\\}\subset{\mathbb{R}^{n}}$,
$\\{\mathpzc{x}_{k}\\}\subset\textnormal{L}^{\infty}({T},{\mathbb{R}^{n}})$,
$\\{\mathpzc{x}_{k}^{\ast}\\}\subset{\textnormal{L}}^{1}({T},{\mathbb{R}^{n}})$,
$\\{\mathpzc{y}_{k}\\}\subset\textnormal{L}^{1}(T,\mathbb{R}^{m})$, and
$\\{\mathpzc{y}_{k}^{\ast}\\}\subset\textnormal{L}^{\infty}(T,\mathbb{R}^{m})$
such that the following assertions hold:

  1. (i)

$\mathpzc{x}_{k}^{*}(t)\in\widehat{D}^{\ast}\Phi_{t}\big{(}\mathpzc{x}_{k}(t),\mathpzc{y}_{k}(t)\big{)}\big{(}\mathpzc{y}_{k}^{\ast}(t)\big{)}$
for a.e. $t\in T$ and all $k\in\mathbb{N}$.

  2. (ii)

$\|\bar{x}-x_{k}\|\to 0$, $\|\bar{x}-\mathpzc{x}_{k}\|_{\infty}\to 0$,
$\displaystyle\int_{T}\|\bar{\mathpzc{y}}(t)-\mathpzc{y}_{k}(t)\|d\mu\to 0$,
and $\|\mathpzc{y}_{k}^{\ast}-\bar{y}^{\ast}\|_{\infty}\to 0$ as $k\to\infty$.

  3. (iii)

$\displaystyle\int_{T}\|\mathpzc{x}_{k}^{*}(t)\|\cdot\|\mathpzc{x}_{k}(t)-x_{k}\|d\mu\to
0$ and $\displaystyle\int_{T}\mathpzc{x}_{k}^{*}(t)d\mu\to\bar{x}^{\ast}$.

Proof. Considering the extended-real-valued function
$\varphi(t,v,w):=\delta_{\operatorname{gph}\Phi_{t}}(u,w)$, observe that
$\varphi$ is a normal integrand. Furthermore, it is easy to see that the
imposed assumptions in (5.2) yield the fulfillment of (3.3) for $\varphi$.
Picking any
$\bar{x}^{\ast}\in\widehat{D}^{\ast}\mathrm{E}_{\Phi}(\bar{x},\bar{y})(\bar{y}^{\ast})$
gives us by regular coderivative definition (2.4) that
$(\bar{x}^{\ast},\bar{y}^{\ast})\in\widehat{\partial}\delta_{\operatorname{gph}\mathrm{E}_{\Phi}}(\bar{x},\bar{y})$.
Now we use the smooth variational description of regular subgradients taken
from (m06, , Theorem 1.88(i)), which tells us that there exist $\eta>0$ and a
function
$\vartheta\colon\mathbb{B}_{\eta}(\bar{x},\bar{y})\to{\mathbb{R}^{n}}\times\mathbb{R}^{m}$
that is Fréchet differentiable at $(\bar{x},\bar{y})$ with
$\nabla\vartheta(\bar{x},\bar{y})=(\bar{x}^{\ast},\bar{y}^{\ast})$ and such
that the difference $\delta_{\operatorname{gph}\mathrm{E}_{\Phi}}-\vartheta$
attains its minimum at $(\bar{x},\bar{y})$ on
$\mathbb{B}_{\eta}(\bar{x},\bar{y})$. Observe that

$\displaystyle\delta_{\operatorname{gph}\mathrm{E}_{\Phi}}\bigg{(}u,\int_{T}\bar{\mathpzc{y}}(t)d\mu\bigg{)}$
$\displaystyle=\int_{T}\varphi_{t}\big{(}u,\bar{\mathpzc{y}}(t)\big{)}d\mu\;\mbox{
and}$
$\displaystyle\delta_{\operatorname{gph}\mathrm{E}_{\Phi}}\bigg{(}u,\int_{T}\mathpzc{w}(t)d\mu\bigg{)}$
$\displaystyle\leq\int_{T}\varphi_{t}\big{(}u,\mathpzc{w}(t)\big{)}d\mu\;\text{
for all
}\;(u,\mathpzc{w})\in{\mathbb{R}^{n}}\times\textnormal{L}^{1}(T,\mathbb{R}^{m}).$

Consider further the extended-real-valued function

${\mathbb{R}^{n}}\times\textnormal{L}^{1}(T,\mathbb{R}^{m})\ni(u,\mathpzc{w})\to\mathrm{E}_{\varphi}(u,\mathpzc{w})-\psi(u,\mathpzc{w})\;\mbox{
with
}\;\psi(u,\mathpzc{w}):=\vartheta\bigg{(}u,\int_{T}\mathpzc{w}(t)d\mu\bigg{)},$

which attains a local minimum at $(\bar{x},\bar{\mathpzc{y}})$. Hence we have
by the elementary version of the subdifferential Fermat rule (see, e.g., (m06,
, Proposition 1.114)) that
$(0,0)\in\widehat{\partial}(\mathrm{E}_{\varphi}-\psi)(\bar{x},\bar{\mathpzc{y}})$.
Observe that $\psi$ is clearly Fréchet differentiable at
$(\bar{x},\bar{\mathpzc{y}})$ with
$\nabla\psi(\bar{x},\bar{\mathpzc{y}})=(\bar{x}^{\ast},\bar{\mathpzc{y}}^{\ast})$,
where $\bar{\mathpzc{y}}^{\ast}$ is the constant function
$\bar{\mathpzc{y}}^{\ast}(t):=\bar{y}^{\ast}\operatorname{\mathds{1}}_{T}(t)$.
Employing now the sum rule from (m06, , Proposition 1.107)) yields
$(\bar{x}^{\ast},\bar{\mathpzc{y}}^{\ast})\in\widehat{\partial}\mathrm{E}_{\varphi}(\bar{x},\bar{\mathpzc{y}})$.
Furthermore, it follows from (5.2) that the function

$t\to\inf_{\mathbb{B}_{\rho}(\bar{x})\times\mathbb{R}^{m}}\big{\\{}\varphi_{t}(\cdot,\cdot)-\langle\bar{\mathpzc{y}}^{\ast}(t),\cdot\rangle\big{\\}}$

is integrable on $T$. Finally, the application of Theorem 3.2 completes the
proof of the proposition. $\hfill\square$

The subsequent results of this section establish pointwise versions of
coderivative Leibniz rules under the integrable quasi-Lipschitzian property of
set-valued integrands $\Phi_{t}$. The first theorem gives us an upper estimate
of the regular coderivative of $\mathrm{E}_{\Phi}$ via the integral of the
limiting coderivative of $\Phi_{t}$.

######  Theorem 5.1 (pointwise estimate for regular coderivatives of expected-
integral mappings)

Let $\Phi\colon T\times{\mathbb{R}^{n}}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ be a set-valued normal integrand
on a complete finite measure space $(T,\mathcal{A},\mu)$, let
$\bar{x}\in\mbox{\rm dom}\,\mathrm{E}_{\Phi}$ satisfy the conditions in (5.2),
and let $\bar{\mathpzc{y}}\in\mathcal{S}_{\Phi}(\bar{x},\bar{y})$ for some
$\bar{y}\in\mathrm{E}_{\Phi}(\bar{x})$. Assume in addition that the integrable
quasi-Lipschitzian property holds for $\Phi$ around
$(\bar{x},\bar{\mathpzc{y}})$. Then we have the inclusion

$\widehat{D}^{\ast}\mathrm{E}_{\Phi}(\bar{x},\bar{y})(y^{\ast})\subset\int_{T}{D}^{\ast}\Phi_{t}\big{(}\bar{x},\bar{\mathpzc{y}}(t)\big{)}(y^{\ast})d\mu\;\mbox{
for all }\;y^{\ast}\in\mathbb{R}^{m}.$ (5.3)

Proof. Pick any
$\bar{x}^{\ast}\in\widehat{D}^{\ast}\mathrm{E}_{\Phi}(\bar{x},\bar{y})(\bar{y}^{\ast})$.
Proposition 7 allows us to find sequences
$\\{x_{k}\\}\subset{\mathbb{R}^{n}}$,
$\\{\mathpzc{x}_{k}\\}\subset\textnormal{L}^{\infty}({T},{\mathbb{R}^{n}})$,
$\\{\mathpzc{x}_{k}^{\ast}\\}\subset{\textnormal{L}}^{1}({T},{\mathbb{R}^{n}})$,
$\\{\mathpzc{y}_{k}\\}\subset\textnormal{L}^{1}(T,\mathbb{R}^{m})$, and
$\\{\mathpzc{y}_{k}^{\ast}\\}\subset\textnormal{L}^{\infty}(T,\mathbb{R}^{m})$
satisfying the conditions
$\mathpzc{x}_{k}^{*}(t)\in\widehat{D}^{\ast}\Phi_{t}(\mathpzc{x}_{k}(t),\mathpzc{y}_{k}(t))(\mathpzc{y}_{k}^{\ast}(t))$
for a.e. $t\in T$, $\|\bar{x}-\mathpzc{x}_{k}\|_{\infty}\to 0$,
$\|\mathpzc{y}_{k}^{\ast}-\bar{y}^{\ast}\|_{\infty}\to 0$, as well as

$\int_{T}\big{\|}\bar{\mathpzc{y}}(t)-\mathpzc{y}_{k}(t)\big{\|}d\mu\to
0\;\mbox{ and
}\;\bigg{\|}\int_{T}\mathpzc{x}_{k}^{*}(t)d\mu-\bar{x}^{\ast}\bigg{\|}\to
0\;\mbox{ when }\;k\to\infty.$

Passing to a subsequence if necessary, we get that that
$(\mathpzc{x}_{k}(t),\mathpzc{y}_{k}(t),\mathpzc{y}_{k}^{\ast}(t))\to(\bar{x},\bar{\mathpzc{y}}(t),\bar{y}^{\ast})$
as $k\to\infty$ for a.e. $t\in T$, which implies by using the limiting
coderivative representation (2.7) that

$\mathop{{\rm Lim}\,{\rm
sup}}_{k\to\infty}\big{\\{}\mathpzc{x}_{k}^{*}(t)\big{\\}}\subset
D^{\ast}\Phi_{t}\big{(}\bar{x},\bar{\mathpzc{y}}(t)\big{)}(\bar{y}^{\ast})\;\mbox{
for a.e. }\;t\in T.$ (5.4)

Employing now the imposed integrable quasi-Lipschitzian property of $\Phi$
around $(\bar{x},\bar{\mathpzc{y}})$ yields

$\|\mathpzc{x}_{k}^{\ast}(t)\|\leq\ell(t)\|\mathpzc{y}_{k}^{\ast}(t)\|\leq\ell(t)M\;\text{
for a.e. }\;t\in T\;\mbox{ and large }\;k\in\mathbb{N},$

where $M:=\sup\\{\|\mathpzc{y}_{k}^{\ast}\|_{\infty}\mbox{ over
}k\in\mathbb{N}\\}$. Then Fatou’s lemma for multifunctions taken from (bs, ,
Corollary 4.1) and being combined with (5.4) tells us that

$\displaystyle\bar{x}^{\ast}\in\mathop{{\rm Lim}\,{\rm
sup}}_{k\to\infty}\left\\{\displaystyle\int_{{T}}\mathpzc{x}_{k}^{*}(t)d\mu\right\\}$
$\displaystyle\subset\displaystyle\int_{{T}}\mathop{{\rm Lim}\,{\rm
sup}}_{k\to\infty}\big{\\{}\mathpzc{x}_{k}^{*}(t)\big{\\}}d\mu$
$\displaystyle\subset\displaystyle\int_{{T}}D^{\ast}\Phi_{t}\big{(}\bar{x},\bar{\mathpzc{y}}(t)\big{)}(y^{\ast})d\mu$

for all $y^{*}\in\mathbb{R}^{m}$. This shows that
$\bar{x}^{\ast}\in\displaystyle\int_{{T}}D^{\ast}\Phi_{t}\big{(}\bar{x},\bar{\mathpzc{y}}(t)\big{)}(y^{\ast})d\mu$
and thus verifies (5.3). $\hfill\square$

To proceed further with deriving an efficient pointwise upper estimate of the
limiting coderivative of $\mathrm{E}_{\Phi}$ via the limiting coderivative of
the integrand $\Phi_{t}$, we need to invoke an additional property of the set-
valued mapping $\mathcal{S}_{\Phi}$ defined in (4.16). This property of
multifunctions, known as inner semicompactness, is formulated and discussed in
(m06, , Definition 1.63(ii)). The reader can find in (m06, , Definition
1.63(i)) a parallel inner semicontinuity property of multifunctions, which
could also be used to establish complemented coderivative Leibniz rules, while
we are not going to pursue this aim in this paper.

Recall that $\mathcal{S}_{\Phi}$ is inner semicompact at $(\bar{x},\bar{y})$
if for every sequence $(x_{k},y_{k})\to(\bar{x},\bar{y})$ there exists a
sequence $\mathpzc{y}_{k}\in\mathcal{S}_{\Phi}(x_{k},y_{k})$ that contains an
$\textnormal{L}^{1}({T},\mathbb{R}^{m})$-norm convergent subsequence as
$k\to\infty$.

######  Theorem 5.2 (coderivative Leibniz rule for integrably quasi-
Lipschitzian multifunctions)

Let $\Phi\colon T\times{\mathbb{R}^{n}}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ be a set-valued normal integrand
on a complete finite measure space $(T,\mathcal{A},\mu)$, and let
$\bar{x}\in\mbox{\rm dom}\,\mathrm{E}_{\Phi}$ satisfy the conditions in (5.2).
Take $\bar{y}\in\mathrm{E}_{\Phi}(\bar{x})$ such that the mapping
$\mathcal{S}_{\Phi}$ is inner semicompact at $(\bar{x},\bar{y})$ and assume in
addition that $\Phi$ is integrably quasi-Lipschitzian around
$(\bar{x},\bar{\mathpzc{y}})$ for all
$\bar{\mathpzc{y}}\in\mathcal{S}_{\Phi}(\bar{x},\bar{y})$. Then we have the
limiting coderivative Leibniz rule

${D}^{\ast}\mathrm{E}_{\Phi}(\bar{x},\bar{y})(y^{\ast})\subset\bigcup\limits_{\bar{\mathpzc{y}}\in\mathcal{S}_{\Phi}(\bar{x},\bar{y})}\int_{T}{D}^{\ast}\Phi_{t}\big{(}\bar{x},\bar{\mathpzc{y}}(t)\big{)}(y^{\ast})d\mu\;\mbox{
whenever }\;y^{\ast}\in\mathbb{R}^{m}.$ (5.5)

Proof. Pick any
$\bar{x}^{\ast}\in{D}^{\ast}\mathrm{E}_{\Phi}(\bar{x},\bar{y})(\bar{y}^{\ast})$.
It follows from representation (2.7) of the limiting coderivative that there
exist the convergent quadruples
$(x_{k},y_{k},x_{k}^{\ast},y_{k}^{\ast})\to(\bar{x},\bar{y},\bar{x}^{\ast},\bar{y}^{\ast})$
with
$x_{k}^{\ast}\in\widehat{D}^{\ast}\mathrm{E}_{\Phi}(x_{k},y_{k})(y_{k}^{\ast})$
for all $k\in\mathbb{N}$. Employing the imposed inner semicompactness of
$\mathcal{S}_{\Phi}$ and passing to a subsequence if necessary give us
$\mathpzc{y}_{k}\in\mathcal{S}_{\Phi}(x_{k},y_{k})$ such that
$\mathpzc{y}_{k}\to\bar{\mathpzc{y}}$ in the norm topology of
$\textnormal{L}^{1}({T},\mathbb{R}^{m})$. Remembering that the graph of
$\Phi_{t}$ is closed yields
$\bar{\mathpzc{y}}\in\mathcal{S}_{\Phi}(\bar{x},\bar{y})$. Applying further
Proposition 7 to each quadruple $(x_{k},y_{k},x_{k}^{\ast},y_{k}^{\ast})$ and
employing the diagonal process ensure the existence of sequences
$\\{\mathpzc{x}_{k}\\}\subset\textnormal{L}^{\infty}({T},{\mathbb{R}^{n}})$,
$\\{\mathpzc{x}_{k}^{\ast}\\}\subset{\textnormal{L}}^{1}({T},{\mathbb{R}^{n}})$,
$\\{\mathpzc{y}_{k}\\}\in\textnormal{L}^{1}(T,\mathbb{R}^{m})$, and
$\\{\mathpzc{y}_{k}^{\ast}\\}\in\textnormal{L}^{\infty}(T,\mathbb{R}^{m})$
satisfying the following conditions:
$\mathpzc{x}_{k}^{*}(t)\in\widehat{D}^{\ast}\Phi_{t}(\mathpzc{x}_{k}(t),\mathpzc{y}_{k}(t))(\mathpzc{y}_{k}^{\ast}(t))$
for a.e. $t\in T$ and all $k\in\mathbb{N}$ together with the norm convergence
$\|\bar{x}-\mathpzc{x}_{k}\|_{\infty}\to 0$,
$\|\bar{\mathpzc{y}}-\mathpzc{y}_{k}\|_{1}\to 0$, and
$\|\mathpzc{y}_{k}^{\ast}-\bar{y}^{\ast}\|_{\infty}\to 0$ as well as

$\bigg{\|}\int_{T}\mathpzc{x}_{k}^{*}(t)d\mu-\bar{x}^{\ast}\bigg{\|}\to
0\;\mbox{ and
}\;\big{(}\mathpzc{x}_{k}(t),\mathpzc{y}_{k}(t),\mathpzc{y}_{k}^{\ast}(t)\big{)}\to\big{(}\bar{x},\bar{\mathpzc{y}}(t),\bar{y}^{\ast}\big{)}\;\mbox{
for a.e. }\;t\in T\;\mbox{ as }\;k\to\infty.$

Furthermore, we have the limiting inclusion (5.4) as obtained in the proof of
Theorem 5.1. Using now the assumed integrable quasi-Lipschitzian property of
$\Phi$ around $(\bar{x},\bar{\mathpzc{y}})$ and remembering the
${\textnormal{L}}^{\infty}$-norm convergence
$\mathpzc{y}_{k}\to\bar{\mathpzc{y}}$ as $k\to\infty$ allow us to find
$\ell\in\textnormal{L}^{1}(T,\mathbb{R}_{+})$ and $k_{0}\in\mathbb{N}$ such
that

$\|\mathpzc{x}_{k}^{\ast}(t)\|\leq\ell(t)\|\mathpzc{y}_{k}^{\ast}(t)\|\leq\ell(t)M\;\mbox{
for almost all }\;t\in T\;\mbox{ and all }\;k\geq k_{0},$

where $M=\sup\\{\|\mathpzc{y}_{k}\|_{\infty}\;\mbox{over}\;k\in\mathbb{N}\\}$.
Hence the functions $\mathpzc{x}_{k}$ are uniformly integrable on $T$. It
follows from (5.4) and the aforementioned Fatou’s lemma for multifunctions
that for all $y^{*}\in\mathbb{R}^{m}$ we get

$\displaystyle\bar{x}^{\ast}\in\displaystyle\int_{{T}}D^{\ast}\Phi_{t}\big{(}\bar{x},\bar{\mathpzc{y}}(t)\big{)}(y^{\ast})d\mu\subset\bigcup\limits_{\bar{\mathpzc{y}}\in\mathcal{S}_{\Phi}(\bar{x},\bar{y})}\int_{T}{D}^{\ast}\Phi_{t}\big{(}\bar{x},\bar{\mathpzc{y}}(t)\big{)}(y^{\ast})d\mu,$

which verifies (5.5) and completes the proof of the theorem. $\hfill\square$

The following result corresponds to an alternative Leibniz rule for
coderivative of expected multifunctions when the mapping $\mathcal{S}$ is not
inner semicompact at the reference point, but the mapping $\Phi$ is integrably
locally Lipschitzian.

######  Theorem 5.3 (coderivative Leibniz rule for integrably Lipschitzian
multifunctions)

Let $\Phi\colon T\times{\mathbb{R}^{n}}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ be a set-valued normal integrand
on a complete finite measure space $(T,\mathcal{A},\mu)$, and let
$\bar{x}\in\mbox{\rm dom}\,\mathrm{E}_{\Phi}$. Suppose that $\Phi$ is
integrably locally Lipschitzian around $\bar{x}$. Then for every
$\bar{y}\in\mathrm{E}_{\Phi}(\bar{x})$ we have the limiting coderivative
Leibniz rule

${D}^{\ast}\mathrm{E}_{\Phi}(\bar{x},\bar{y})(y^{\ast})\subset\int_{T}{D}^{\ast}\Phi_{t}\big{(}\bar{x},\Phi_{t}(\bar{x})\big{)}(y^{\ast})d\mu\;\mbox{
whenever }\;y^{\ast}\in\mathbb{R}^{m},$ (5.6)

where
${D}^{\ast}\Phi_{t}\big{(}\bar{x},\Phi_{t}(\bar{x})\big{)}(y^{\ast}):=\bigcup_{y\in\Phi_{t}(\bar{x})}{D}^{\ast}\Phi_{t}\big{(}\bar{x},y)(y^{\ast})$.

###### Proof

Following the arguments of Theorem 5.2, we have the existence of sequences
$\\{\mathpzc{x}_{k}\\}\subset\textnormal{L}^{\infty}({T},{\mathbb{R}^{n}})$,
$\\{\mathpzc{x}_{k}^{\ast}\\}\subset{\textnormal{L}}^{1}({T},{\mathbb{R}^{n}})$,
$\\{\mathpzc{y}_{k}\\}\in\textnormal{L}^{1}(T,\mathbb{R}^{m})$, and
$\\{\mathpzc{y}_{k}^{\ast}\\}\in\textnormal{L}^{\infty}(T,\mathbb{R}^{m})$
satisfying the following conditions:
$\mathpzc{x}_{k}^{*}(t)\in\widehat{D}^{\ast}\Phi_{t}(\mathpzc{x}_{k}(t),\mathpzc{y}_{k}(t))(\mathpzc{y}_{k}^{\ast}(t))$
for a.e. $t\in T$ and all $k\in\mathbb{N}$ together with the norm convergence
$\|\bar{x}-\mathpzc{x}_{k}\|_{\infty}\to 0$, and
$\|\mathpzc{y}_{k}^{\ast}-\bar{y}^{\ast}\|_{\infty}\to 0$ as well as
$\|\int_{T}\mathpzc{x}_{k}^{*}(t)d\mu-\bar{x}^{\ast}\|\to 0$ and
$\|\mathpzc{x}_{k}^{\ast}(t)\|\leq\ell(t)M$ for a.e. $t\in T$. Now, by Fatou’s
lemma for multifunctions we get that

$\displaystyle x^{\ast}\in\mathop{{\rm Lim}\,{\rm
sup}}_{k\to\infty}\left\\{\displaystyle\int_{{T}}\mathpzc{x}_{k}^{*}(t)d\mu\right\\}$
$\displaystyle\subset\displaystyle\int_{{T}}\mathop{{\rm Lim}\,{\rm
sup}}_{k\to\infty}\big{\\{}\mathpzc{x}_{k}^{*}(t)\big{\\}}d\mu.$

To conclude the proof, let us show the following inclusion $\mathop{{\rm
Lim}\,{\rm
sup}}_{k\to\infty}\\{\mathpzc{x}_{k}^{*}(t)\\}\subset{D}^{\ast}\Phi_{t}\big{(}\bar{x},\Phi_{t}(\bar{x})\big{)}(y^{\ast})$
for almost all $t\in T$. Indeed, consider a set of full measure $\hat{T}$,
where $\mathpzc{x}_{k}(t)\to\bar{x}$ and $\mathpzc{y}_{k}(t)\to\bar{y}$. Fix
$t\in\hat{T}$ and $u^{\ast}\in\mathop{{\rm Lim}\,{\rm
sup}}_{k\to\infty}\\{\mathpzc{x}_{k}^{*}(t)\\}$. Hence, by definition there
exists a subsequence $x^{\ast}_{k_{j}}(t)\to u^{\ast}$. Since
$\|\mathpzc{y}_{k_{j}}(t)\|\leq\kappa(t)$ (recall (5.2)), we can take assume
that $\mathpzc{y}_{k_{j}}\to y$ for some $y\in\Phi_{t}(\bar{x})$ (recall that
the graph of $\Phi_{t}$ is closed). Therefore, by the limiting coderivative
representation (2.7), we get that
$x^{\ast}\in\limsup_{j\to\infty}\widehat{D}^{\ast}\Phi_{t}(\mathpzc{x}_{k_{j}}(t),\mathpzc{y}_{k_{j}}(t))(\mathpzc{y}_{k_{j}}^{\ast}(t))\subset{D}^{\ast}\Phi_{t}\big{(}\bar{x},y)\big{)}(y^{\ast})$,
and that concludes the proof.

Next we present a simple consequence of Theorem 5.2 that is used in what
follows.

######  Corollary 1 (coderivative Leibniz rule for single-valued expected-
integral mappings)

Under the general assumptions of Theorem 5.2, suppose in addition that
$\mathrm{E}_{\Phi}(\bar{x})$ is a singleton, and that $\Phi$ is integrably
locally Lipschitzian around $\bar{x}$. Then the mapping $\mathcal{S}_{\Phi}$
is inner semicompact at $(\bar{x},\mathrm{E}_{\Phi}(\bar{x}))$ and we have the
coderivative Leibniz rule

${D}^{\ast}\mathrm{E}_{\Phi}(\bar{x})(y^{\ast})\subset\int_{T}{D}^{\ast}\Phi_{t}\big{(}\bar{x}\big{)}(y^{\ast})d\mu\;\mbox{
for all }\;y^{*}\in\mathbb{R}^{m}.$ (5.7)

Proof. Denote $\bar{y}:=\mathrm{E}_{\Phi}(\bar{x})$ and take a measurable
selection $\bar{\mathpzc{y}}(t)\in\Phi_{t}(\bar{x})$ for a.e. $t\in T$. To
verify that $\mathcal{S}_{\Phi}$ is inner semicompact at $(\bar{x},\bar{y})$,
let $(x_{k},y_{k})\to(\bar{x},\bar{y})$ as $k\to\infty$, and let
$\mathpzc{y}_{k}\in\mathcal{S}_{\Phi}(x_{k},y_{k})$ for all $k\in\mathbb{N}$.
Employing the integrable local Lipschitzian property (4.2) of $\Phi$ around
$\bar{x}$ tells us that

$\displaystyle\|\mathpzc{y}_{k}(t)-\bar{\mathpzc{y}}(t)\|\leq\ell(t)\|x_{k}-\bar{x}\|\;\text{
for a.e. }\;t\in T\;\mbox{ and all large }\;k\in\mathbb{N}.$

Then it follows from Lebesgue’s dominated convergence theorem that
$\mathpzc{y}_{k}\to\bar{\mathpzc{y}}$ in
$\textnormal{L}^{1}(T,\mathbb{R}^{m})$ as $k\to\infty$, which verifies the
inner semicompactness property of $\mathcal{S}_{\Phi}$ at $(\bar{x},\bar{y})$.
Applying finally Theorem 5.2, we arrive at (5.7) and thus complete the proof
of the corollary. $\hfill\square$

As we see, the previous versions of the coderivative Leibniz rule provided
just upper estimates of the regular and limiting coderivatives of the
expected-integral mappings. Although calculus rules as inclusions of this type
suffice for many applications, it is important to establish efficient
conditions ensuring the fulfillment of such results as equalities. The
following theorem does the job. We say that a set-valued mapping $F$ is
coderivative regular at $(\bar{x},\bar{y})\in\operatorname{gph}F$ for
$y^{\ast}\in\mathbb{R}^{m}$ if
$D^{\ast}F(\bar{x},\bar{y})(y^{\ast})=\widehat{D}^{\ast}F(\bar{x},\bar{y})(y^{\ast})$.

######  Theorem 5.4 (coderivative Leibniz rule as equality)

In the setting of Corollary 1, suppose that for a.e. $t\in T$ we have that
$\Phi_{t}(\bar{x})=\\{\bar{\mathpzc{y}}(t)\\}$ and that $\Phi_{t}$ is
coderivative regular at $(\bar{x},\bar{\mathpzc{y}}(t))$ for
$y^{\ast}\in\mathbb{R}^{m}$. Then

$\widehat{D}^{\ast}\mathrm{E}_{\Phi}(\bar{x})(y^{\ast})={D}^{\ast}\mathrm{E}_{\Phi}(\bar{x})(y^{\ast})=\int_{T}{D}^{\ast}\Phi_{t}\big{(}\bar{x}\big{)}(y^{\ast})d\mu.$
(5.8)

###### Proof

Taking into account the inclusion of Theorem 5.2, it is sufficient to show
that

$\int_{T}{D}^{\ast}\Phi_{t}\big{(}\bar{x},\bar{\mathpzc{y}}(t)\big{)}(y^{\ast})d\mu\subset\widehat{D}^{\ast}\mathrm{E}_{\Phi}(\bar{x})(y^{\ast}).$
(5.9)

First we check that for all $(u,v)\in{\mathbb{R}^{n}}\times\mathbb{R}^{m}$
there exists $\mathpzc{v}\in\textnormal{L}^{1}(T,\mathbb{R}^{m})$ with
$v=\int_{T}\mathpzc{v}(t)d\mu$ such that

$\displaystyle\int_{T}\overline{\mbox{\rm
co}\,}d_{\operatorname{gph}\Phi_{t}}(\bar{x},\bar{\mathpzc{y}}(t))(u,\mathpzc{v}(t))d\mu\leq
d_{\operatorname{gph}\mathrm{E}_{\Phi}}(\bar{x},\bar{y})(u,v),$ (5.10)

where $d_{\operatorname{gph}\mathrm{E}_{\Phi}}$ denotes the subderivative
(2.11) of the indicator function of $\operatorname{gph}\mathrm{E}_{\Phi}$, and
where $\overline{\mbox{\rm co}\,}d_{\operatorname{gph}\Phi_{t}}$ is the convex
closure of this subderivative. This claim is obvious if right-hand side of
(5.10) is $\infty$. Otherwise, suppose that
$d_{\operatorname{gph}\mathrm{E}_{\Phi}}(\bar{x},\bar{y})(u,v)<\infty$, we
find sequences $s_{k}\downarrow 0$ and $(u_{k},v_{k})\to 0$ as $k\to\infty$
such that

$\displaystyle
d_{\operatorname{gph}\mathrm{E}_{\Phi}}(\bar{x},\bar{y})(u,v)=\lim\limits_{k\to\infty}\frac{\delta_{\operatorname{gph}\mathrm{E}_{\Phi}}(\bar{x}+s_{k}u_{k},\bar{y}_{k}+s_{k}v_{k})}{s_{k}}.$

The above inequality implies that whenever $k\in\mathbb{N}$ is sufficiently
large there exists $\mathpzc{w}_{k}(t)\in\Phi_{t}(\bar{x}+s_{k}u_{k})$ for
a.e. $t\in T$ such that $\int_{T}\mathpzc{w}_{k}(t)d\mu=\bar{y}+s_{k}v_{k}$.
By the integrable Lipschitz continuity of $\Phi$, it can be assumed that for
a.e. $t\in T$ and all $k\in\mathbb{N}$ we have the representation

$\displaystyle\mathpzc{w}_{k}(t)=\bar{\mathpzc{y}}(t)+s_{k}\mathpzc{v}_{k}(t)\;\text{
with some measurable function
}\;\mathpzc{v}_{k}(t)\in\|u_{k}\|\ell(t)\mathbb{B},$

which tells us that
$\|\mathpzc{v}_{k}(t)\|\leq\max\big{\\{}\|u_{k}\|\;\big{|}\;k\in\mathbb{N}\\}\ell(t)$.
This implies that a subsequence of $\\{\mathpzc{v}_{k}\\}$ converges weakly to
some $\mathpzc{v}\in\textnormal{L}^{1}(T,\mathbb{R}^{m})$. Thus
$v_{k}=\int_{T}\mathpzc{v}_{k}(t)d\mu(t)\to\int_{T}\mathpzc{v}(t)d\mu$ along
this subsequence, and therefore we arrive at $v=\int_{T}\mathpzc{v}(t)d\mu$.

Consider further the integrand $\Psi\colon
T\times[0,\infty]\times{\mathbb{R}^{n}}\times\mathbb{R}^{m}\to\overline{\mathbb{R}}$
defined by

$\displaystyle\Psi(t,r,a,b):=\left\\{\begin{array}[]{cc}\displaystyle\frac{\delta_{\operatorname{gph}\Phi_{t}}(\bar{x}+ra,\bar{\mathpzc{y}}(t)+rb)}{r}&\text{
if }r>0,\\\ &\\\ \overline{\mbox{\rm
co}\,}d_{\operatorname{gph}\Phi_{t}}(\bar{x},\bar{\mathpzc{y}}(t))(a,b)&\text{
if }r=0.\end{array}\right.$ (5.14)

It is easy to see that $\Psi(t,\cdot)$ is lower semicontinuous for all $t\in
T$, and also that for every $(t,r,a)\in
T_{na}\times[0,\infty)\times{\mathbb{R}^{n}}$ the function $\Psi(t,r,a,\cdot)$
is convex. Applying (bal, , Theorem 2.1) on $T_{na}$, we get that

$\displaystyle\int_{T_{pa}}\overline{\mbox{\rm
co}\,}d_{\operatorname{gph}\Phi_{t}}\big{(}\bar{x},\bar{\mathpzc{y}}(t)\big{)}\big{(}u,\mathpzc{v}(t)\big{)}d\mu$
$\displaystyle=\int_{T_{na}}\Psi\big{(}t,0,u,\mathpzc{v}(t)\big{)}d\mu\leq\liminf\limits_{k\to\infty}\int_{T_{na}}\Psi\big{(}t,s_{k},u_{k},\mathpzc{v}_{k}(t)\big{)}d\mu=0.$

On the other hand, noticing that $\mathpzc{v}_{k}$ converges pointwise to
$\mathpzc{v}$ on $T_{pa}$, we deduce from Fatou’s lemma and the lower
semicontinuity of $\Psi$ that

$\displaystyle\int_{T_{pa}}\overline{\mbox{\rm
co}\,}d_{\operatorname{gph}\Phi_{t}}\big{(}\bar{x},\bar{\mathpzc{y}}(t)\big{)}\big{(}u,\mathpzc{v}(t)\big{)}d\mu$
$\displaystyle=\int_{T_{pa}}\Psi\big{(}t,0,u,\mathpzc{v}(t)\big{)}d\mu\leq\int_{T_{na}}\liminf\limits_{k\to\infty}\Psi\big{(}t,s_{k},u_{k},\mathpzc{v}_{k}(t)\big{)}$
$\displaystyle\leq\liminf\limits_{k\to\infty}\int_{T_{na}}\Psi\big{(}t,s_{k},u_{k},\mathpzc{v}_{k}(t)\big{)}d\mu=0,$

which therefore verifies the estimate in (5.10). To proceed, pick
$x^{\ast}\in\int_{T}{D}^{\ast}\Phi_{t}\big{(}\bar{x},\bar{\mathpzc{y}}(t)\big{)}(y^{\ast})d\mu$
and then find, by using the assumed graphical regularity of the integrand
$\Phi_{t}$, a measurable selection
$\mathpzc{x}(t)\in{D}^{\ast}\Phi_{t}\big{(}\bar{x},\bar{\mathpzc{y}}(t)\big{)}(y^{\ast})=\widehat{D}^{\ast}\Phi_{t}\big{(}\bar{x},\bar{\mathpzc{y}}(t)\big{)}(y^{\ast})$
such that $x^{\ast}=\int_{T}\mathpzc{x}^{\ast}(t)d\mu$. This implies due to
(2.10) that

$\displaystyle\langle\mathpzc{x}^{\ast}(t),u\rangle-\langle
y^{\ast},v\rangle\leq
d_{\operatorname{gph}\Phi_{t}}\big{(}\bar{x},\bar{\mathpzc{y}}(t)\big{)}(u,u)\;\text{
for all }\;(u,v)\in{\mathbb{R}^{n}}\times\mathbb{R}^{m},$

which ensures by definition of the convex closure the estimate

$\displaystyle\langle\mathpzc{x}^{\ast}(t),u\rangle-\langle
y^{\ast},v\rangle\leq\overline{\mbox{\rm
co}\,}d_{\operatorname{gph}\Phi_{t}}\big{(}\bar{x},\bar{\mathpzc{y}}(t)\big{)}(u,v)\;\text{
for all }\;(u,v)\in{\mathbb{R}^{n}}\times\mathbb{R}^{m}.$

Fixing $(u,v)\in{\mathbb{R}^{n}}\times\mathbb{R}^{m}$, find $\mathpzc{v}$
satisfying (5.10), and so

$\displaystyle\langle x^{\ast},u\rangle-\langle y^{\ast},v\rangle$
$\displaystyle=\int_{{T}}\left(\langle\mathpzc{x}^{\ast}(t),u\rangle-\langle
y^{\ast},\mathpzc{v}(t)\rangle\right)d\mu$
$\displaystyle\leq\int_{T}\overline{\mbox{\rm
co}\,}d_{\operatorname{gph}\Phi_{t}}\big{(}\bar{x},\bar{\mathpzc{y}}(t)\big{)}\big{(}u,\mathpzc{v}(t)\big{)}d\mu$
$\displaystyle\leq
d_{\operatorname{gph}\mathrm{E}_{\Phi}}(\bar{x},\bar{y})(u,v).$

Invoking now (2.10) yields
$x^{\ast}\in\widehat{D}^{\ast}\mathrm{E}_{\Phi}\bar{x},\bar{y})(y^{\ast})$,
which verifies (5.9) and thus completes the proof.

The coderivative Leibniz rules obtained above in terms of the limiting
coderivative and the coderivative characterizations of Lipschitzian properties
of deterministic multifunctions discussed in Section 4 allow us to establish
efficient conditions for Lipschitz stability of expected-integral mappings. We
present here the following result ensuring the Lipschitz-like property of the
multifunction $\mathrm{E}_{\Phi}$.

######  Proposition 8 (Lipschitz-like property of expected-integral
multifunctions)

In the setting of Theorem 5.2, assume in addition that

$\int_{T}{D}^{\ast}\Phi_{t}\big{(}\bar{x},\bar{\mathpzc{y}}(t)\big{)}(0)d\mu=\big{\\{}0\big{\\}}\;\mbox{
for all }\;\bar{\mathpzc{y}}\in\mathcal{S}_{\Phi}(\bar{x},\bar{y}).$ (5.15)

Then the expected-integral multifunction $\mathrm{E}_{\Phi}$ from (1.2) is
Lipschitz-like around $(\bar{x},\bar{y})$.

Proof. We readily deduce this statement from the coderivative Leibnitz rule
(5.5) and the coderivative criterion (4.9) for the Lipschitz-like property of
deterministic set-valued mappings. $\hfill\square$

We conclude this section with a direct consequence of the obtained
coderivative Leibniz rules of the inclusion and equality types to derive the
corresponding subdifferential Leibniz rule for expected-integral functionals.
The results in this vein can be found in different settings in chp19 ; mor-
sag18 ; chp20 ; chp192 .

###### Proposition 9

Let $\varphi\colon T\times\mathbb{R}^{n}\to\overline{\mathbb{R}}$ be a normal
integrand. Take $\bar{x}\in{\mathbb{R}^{n}}$ and assume that there exist a
positive integrable function $\ell\colon T\to(0,\infty)$ and a number
$\varepsilon>0$ such that

$\displaystyle|\varphi(t,x)-\varphi(t,y)|\leq\ell(t)\|x-y\|\;\text{ for all
}\;x,y\in\mathbb{B}_{\varepsilon}(\bar{x})\;\mbox{ and a.e. }\;t\in T.$

Then we have the limiting subdifferential Leibniz rule

$\partial\mathrm{E}_{\varphi}(\bar{x})\subset\int_{T}\partial\varphi_{t}(\bar{x})d\mu,$
(5.16)

where $\varphi_{t}(x):=\varphi(t,x)$. Furthermore, (5.16) holds as equality if
$\varphi_{t}$ is lower regular at $\bar{x}$ for a.e. $t\in T$.

Proof. The inclusion result in (5.16) follows directly from Corollary 1. To
verify the equality therein, observe that
$D^{\ast}\varphi(\bar{x})(1)=\partial\varphi(\bar{x})$ for real-valued l.s.c.
functions (see (m18, , Theorem 1.23)), and that we always have
$\widehat{D}^{\ast}\varphi(\bar{x})(1)\supset\widehat{\partial}\varphi(\bar{x})$.
Invoking finally the assumed lower regularity verifies the equality in (5.16)
and thus completes the proof of the proposition. $\hfill\square$

##  6 Lipschitz Stability and Coderivatives of Random Mappings with Composite
Integrands

This section is devoted to the study of expected-integral multifunctions
$\mathrm{E}_{\Phi}$ from (1.2) with $\Phi\colon
T\times{\mathbb{R}^{n}}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ generated by composite set-valued
normal integrands of the type

$\Phi_{t}(x)=F\big{(}t,g_{t}(x)\big{)}\;\text{ for all }\;x\in U\;\text{ and
a.e. }\;t\in T$ (6.1)

near some point $\bar{x}\in\mbox{\rm dom}\,\mathrm{E}_{\Phi}$, where $F\colon
T\times\mathbb{R}^{q}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ and $g\colon T\times
U\to\mathbb{R}^{q}$. First we reveal general conditions on $F$ and $g$
ensuring that $\Phi$ is the integrably locally Lipschitzian in the sense of
(4.2) around the reference point. Then we introduce a new notion of integrable
amenable multifunctions, which expands to (both random and deterministic) set-
valued mappings the fundamental notion of amenable extended-real-valued
deterministic functions that was comprehensively studied in rw . The Leibniz-
type rules obtained for such compositions give us pointwise upper estimates of
both regular and limiting coderivatives of (1.2) via compositions of limiting
coderivatives under the integral sign. Finally, we apply the obtained results
to evaluate coderivatives of expected-integral multifunctions generated by
feasible set mappings in constrained stochastic programming.

Let us start with establishing Lipschitz stability of expected-integral
multifunctions with composite integrals (6.1). The following lemma for
deterministic set-valued mappings with convex graphs and its very simple proof
are of their own interest. This result can be treated as a quantitative
counterpart (without the closed-graph assumption) of (m93, , Theorems 5.9 and
5.12). Recall r  that a set-valued mapping $G\colon{\mathbb{R}^{n}}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ is sub-Lipschitzian around
$\bar{x}\in\mbox{\rm dom}\,G$ if for every compact set
$V\subset\mathbb{R}^{m}$ there exist a neighborhood $U$ of $\bar{x}$ and a
number $\ell\geq 0$ such that we have the inclusion

$G(x)\cap V\subset G(x^{\prime})+\ell\|x-x^{\prime}\|\mathbb{B}\;\mbox{ for
all }\;x,x^{\prime}\in U.$ (6.2)

The coderivative characterization from (m93, , Theorem 5.9(b)) of the sub-
Lipschitzian property (6.2) for (locally) closed-graph multifunctions reads as
follows: $G$ is sub-Lipschitzian around $\bar{x}$ if and only if for any
compact set $V\subset\mathbb{R}^{m}$ there exist $\eta>0$ and $\ell\geq 0$
such that

$\sup\big{\\{}\|x^{*}\|\;\big{|}\;x^{*}\in
D^{*}G(x,y)(y^{*})\big{\\}}\leq\ell\|y^{*}\|\;\mbox{ whenever
}\;y^{*}\in\mathbb{R}^{m}$ (6.3)

for all $x\in\mathbb{B}_{\eta}(\bar{x})$ and $y\in G(x)\cap V$. Here is our
quantitative version for convex-graph multifunctions.

######  Lemma 1 (coderivative estimate for convex-graph multifunctions)

Let $G\colon{\mathbb{R}^{n}}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ be a convex-graph multifunction
with $\bar{x}\in\mbox{\rm dom}\,G$ for which there exist positive numbers $M$
and $\eta$ such that

${\rm dist}\big{(}0;G(x)\big{)}\leq M\;\text{ whenever
}\;x\in\mathbb{B}_{2\eta}(\bar{x}).$ (6.4)

Then for all $x\in\mathbb{B}_{\eta}(\bar{x})$, all $y\in G(x)$, and all
$y^{*}\in\mathbb{R}^{m}$ we have the coderivative estimate

$\sup\big{\\{}\|x^{\ast}\|\;\big{|}\;x^{\ast}\in
D^{\ast}G(x,y)(y^{\ast})\big{\\}}\leq\left(\frac{M+\|y\|}{\eta}\right)\|y^{\ast}\|.$
(6.5)

If in addition the graph of $G$ is locally closed, then this multifunction is
sub-Lipschitzian around $\bar{x}$.

Proof. Pick any $x\in\mathbb{B}_{\eta}(\bar{x})$, $y\in G(x)$, and
$x^{\ast}\in D^{\ast}G(x,y)(y^{\ast})$ with $y^{\ast}\in\mathbb{R}^{m}$. Since
the graph of $G$ is convex and (6.4) holds, we get the inequalities (see,
e.g., (m18, , Proposition 1.7))

$\langle x^{\ast},x+h-x\rangle\leq\langle
y^{\ast},y_{h}-y\rangle\leq\|y^{\ast}\|(M+\|y\|)\;\mbox{ for all
}\;h\in\mathbb{B}_{\eta}(0),$

where $y_{h}\in G(x+h)$ is such that $\|y_{h}\|={\rm dist}(0;G(x+h))$. This
readily yields (6.5). Finally, the sub-Lipschitzian property of $G$ around
$\bar{x}$ follows from (6.3) and (6.5) with $\ell:=(M+\|y\|)/\eta$.
$\hfill\square$

The second lemma is more technical while being used below in the proofs of
both theorems in this section. To proceed, we need the following assumption:
for all $h\in\mathbb{B}_{\eta}(\bar{x})$ the function

$t\mapsto{\rm dist}\big{(}0;F_{t}(g_{t}(\bar{x})+h)\big{)}\;\mbox{ is
integrable on }\;T.$ (6.6)

######  Lemma 2 (perturbed distance estimate)

Let $F\colon T\times\mathbb{R}^{q}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ be a convex normal integrand
satisfying condition (6.6). Then there exist
$\kappa\in\textnormal{L}^{1}(T,\mathbb{R}_{+})$ and $\eta_{1}\in(0,\eta)$ such
that

$\mbox{\rm dist}\,\big{(}0;F_{t}(g_{t}(\bar{x})+h)\big{)}\leq\kappa(t)\;\text{
for all }\;h\in\eta_{1}\mathbb{B}\;\text{ and a.e. }\;t\in T.$ (6.7)

Proof. Take finitely many vectors $e_{i}\in\eta\mathbb{B}$ as $i\in I$ such
that $0\in\mbox{\rm int}\,\Delta$, where $\Delta$ is the simplex generated by
vectors $\\{e_{i}\\}_{i\in I}$, i.e., given by

$\Delta:=\bigg{\\{}\sum_{i\in I}\lambda_{i}e_{i}\;\bigg{|}\;\lambda_{i}\geq
0,\;\sum_{i\in I}\lambda_{i}=1\bigg{\\}}.$

Define the function $\kappa(t):=\max\\{{\rm
dist}(0;F_{t}(g_{t}(\bar{x})+e_{i}))\;|\;i\in I\\}$, which is integrable on
$T$ by assumption (6.6), and then pick any $h$ in the form $h=\sum_{i\in
I}\lambda_{i}e_{i}$. Recalling that the graph of $F_{t}$ is convex for a.e.
$t\in T$ ensures the fulfillment of the estimates

${\rm dist}\big{(}0;F_{t}(g_{t}(\bar{x})+h)\big{)}\leq\sum_{i\in
I}\lambda_{i}{\rm
dist}\big{(}0;F_{t}(g_{t}(\bar{x})+e_{i})\big{)}\leq\kappa(t)\;\mbox{ for a.e.
}\;t\in T.$

Choosing finally $\eta_{1}>0$ with $\eta_{1}\mathbb{B}\subset\Delta$ verifies
(6.7) and thus completes the proof of the lemma. $\hfill\square$

Having Lemmas 1 and 2 in hand, we now establish the integrable local
Lipschitzian property of convex composite normal integrands (6.1) under some
additional assumptions.

######  Theorem 6.1 (integrable local Lipschitzian property of composite
integrands)

Let $\Phi\colon T\times{\mathbb{R}^{n}}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ be a normal integrand represented
in the composite form (6.1) around $\bar{x}\in\mbox{\rm
dom}\,\mathrm{E}_{\Phi}$ on a complete finite measure space $(T,{\cal
A},\mu)$, where $F\colon T\times\mathbb{R}^{q}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ is a set-valued convex normal
integrand that is integrably bounded as in (5.2), and where $g_{t}(x)$ is
continuously differentiable around $\bar{x}$ with the uniformly bounded
gradients $\nabla g_{t}(x)$ for all $x\in\mathbb{B}_{\eta}(\bar{x})\subset U$
and for a.e. $t\in T$ under the fulfillment of (6.6). Then the multifunction
$\Phi$ is integrably locally Lipschitzian around $\bar{x}$.

Proof. Select a set $\widehat{T}\subset T$ of full measure, and then take
$\eta>0$ and $\kappa\in\textnormal{L}^{1}(T,\mathbb{R}_{+})$ such that $F$
satisfies (6.7) for all $t\in\widehat{T}$, and that $\Phi$ satisfies the
integrable bounded condition from (5.2) over $\mathbb{B}_{\eta}(\bar{x})$ on
$\widehat{T}$ with this function $\kappa$. Now choose $\varepsilon\in(0,\eta)$
ensuring the inequalities

$\sup\big{\\{}\|\nabla
g_{t}(x)\|\;\big{|}\;(t,x)\in\widehat{T}\times\mathbb{B}_{\varepsilon}(\bar{x})\big{\\}}\leq\ell\;\mbox{
and}$ $\|g_{t}(x)-g_{t}(u)\|\leq\ell\|x-u\|\;\mbox{
whenever}\;x,u\in\mathbb{B}_{\varepsilon}(\bar{x}),\;t\in\widehat{T}$

for some $\ell>0$. Suppose without loss of generality that
$\varepsilon\ell<\eta$. Now taking $t\in\widehat{T}$,
$x\in\mathbb{B}_{\varepsilon/2}(\bar{x})$, $y\in\Phi_{t}(x)$, and $x^{\ast}\in
D^{\ast}\Phi_{t}(x,y)$, we observe that
$g_{t}(x)\in\mathbb{B}_{\eta/2}(g_{t}(\bar{x}))$. Applying Lemma 1 to the
mapping $G:=F_{t}$ at the point $g(\bar{x})$ gives us the coderivatives
estimate (6.5), which ensures by the coderivative criterion (4.9) that $F_{t}$
is Lipschitz-like around $(g_{t}(x),y)$. Then applying the coderivative chain
rule for deterministic multifunctions from (m18, , Theorem 3.11(iii)) to the
composition in (6.1) yields the representation

$x^{\ast}=\nabla g_{t}(x)^{\ast}\circ z^{\ast}\;\mbox{ with some
}\;z^{\ast}\in D^{\ast}F_{t}\big{(}g(x),y\big{)}(y^{\ast})\;\mbox{ and
}\;y^{*}\in\mathbb{R}^{m}.$

Employing again by the coderivative estimate (6.5) leads us to the inequality

$\|z^{\ast}\|\leq\frac{4\kappa(t)}{\eta}\|y^{\ast}\|,$

where we use that $\|y(t)\|\leq\kappa(t)$ due to the choice of
$y\in\Phi_{t}(x)$. This tells us that

$\|x^{\ast}\|=\|\nabla g_{t}(x)^{\ast}\circ z^{\ast}\|\leq\|\nabla
g_{t}(x)^{\ast})\|\cdot\|z^{\ast}\|\leq\frac{4\ell\kappa(t)}{\eta}\|y^{\ast}\|.$

Since the obtained estimate holds for all $t\in\widehat{T}$,
$x\in\mathbb{B}_{\varepsilon/2}(\bar{x})$, and $y\in\Phi_{t}(x)$, we arrive at
the coderivative condition (4.10), which ensures by Theorem 4.1 that $\Phi$ is
integrably locally Lipschitzian around $\bar{x}$. $\hfill\square$

Next we introduce a new property of random multifunctions that plays a crucial
role in deriving composite coderivative Leibniz rules in what follows.

######  Definition 4 (integrably amenable multifunctions)

A set-valued normal integrand $\Phi\colon
T\times{\mathbb{R}^{n}}\to\mathbb{R}^{q}$ defined on a complete finite measure
space $(T,\mathcal{A},\mu)$ is integrably amenable at
$(\bar{x},\bar{y})\in\operatorname{gph}\mathrm{E}_{\Phi}$ if there exist a
neighborhood $U$ of $\bar{x}$, a set-valued convex normal integrand $F\colon
T\times\mathbb{R}^{q}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$, and a function $g\colon T\times
U\to\mathbb{R}^{q}$, which is measurable with respect to $t$ on $T$ and
continuously differentiable with respect to $x$ around $\bar{x}$, such that
the composite representation (6.1) holds, and that the following qualification
condition is satisfied for a.e. $t\in T$:

$D^{\ast}F_{t}\big{(}g_{t}(\bar{x}),\bar{\mathpzc{y}}(t)\big{)}(0)\cap\operatorname{Ker}\nabla
g_{t}(\bar{x})^{*}=\\{0\\}\;\mbox{ whenever }\;\bar{\mathpzc{y}}\in{\cal
S}_{\Phi}(\bar{x},\bar{y}).$ (6.8)

It follows from Definition 3(ii) that the qualification condition (6.8)
fulfills automatically if the outer mapping $F$ in (6.1) is integrably quasi-
Lipschitzian around $(\bar{x},\bar{\mathpzc{y}})$ for each
$\bar{\mathpzc{y}}\in{\cal S}_{F}(\bar{x},\bar{y})$. Recall that Theorem 4.2
ensures the fulfillment of (6.8) when $F$ is integrably Lipschitz-like around
$(\bar{x},\bar{\mathpzc{y}})$ for each $\bar{\mathpzc{y}}\in{\cal
S}(\bar{x},\bar{y})$, provided that $(T,\mu,\mathcal{A})$ is a purely atomic
measure space consisting of countable disjoint family of atoms. In the
alternative setting of nonatomic spaces, we deduce from Theorem 4.3 that the
integrable quasi-Lipschitzian property of $F$ to ensure (6.8) can be
equivalently replaced but its locally Lipschitzian counterpart. On the other
hand, observe from (6.8) that this condition is satisfied, independently of
the Lipschitzian properties of $F$, if the Jacobian $\nabla g_{t}(\bar{x})$ is
of full rank for a.e. $t\in T$.

Now we are ready to derive the pointwise coderivative Leibniz rules for both
regular and limiting coderivatives of expected-integral multifunctions with
amenable set-valued integrands.

######  Theorem 6.2 (coderivative Leibniz rules for integral multifunctions
with amenable integrands)

Let $\Phi\colon T\times{\mathbb{R}^{n}}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ be an amenable multifunction at
$(\bar{x},\bar{y})\in\operatorname{gph}\mathrm{E}_{\Phi}$, which is assumed to
be integrably bounded as in (5.2) with the uniformly bounded gradients $\nabla
g_{t}(x)$ while satisfying the integrability condition (6.6) for the outer
mapping $F$ in (6.1). Then for every
$\bar{\mathpzc{y}}\in\mathcal{S}_{\Phi}(\bar{x},\bar{y})$ and every
$y^{\ast}\in\mathbb{R}^{m}$ the regular coderivative upper estimate

$\widehat{D}^{\ast}\mathrm{E}_{\Phi}(\bar{x},\bar{y})(y^{\ast})\subset\int_{T}\nabla
g_{t}(\bar{x})^{\ast}\circ
D^{\ast}F_{t}\big{(}g_{t}(\bar{x}),\bar{\mathpzc{y}}(t)\big{)}(y^{\ast})d\mu$
(6.9)

holds. If in addition the multifunction $\mathcal{S}_{\Phi}$ is inner
semicompact at $(\bar{x},\bar{y})$, then we have the following upper estimate
of the limiting coderivative of $\mathrm{E}_{\Phi}$ at $(\bar{x},\bar{y})$:

$D^{\ast}\mathrm{E}_{\Phi}(\bar{x},\bar{y})(y^{\ast})\subset\bigcup\limits_{\bar{\mathpzc{y}}\in\mathcal{S}_{\Phi}(\bar{x},\bar{y})}\int_{T}\nabla
g_{t}(\bar{x})^{\ast}\circ{D}^{\ast}F_{t}\big{(}g_{t}(\bar{x}),\bar{\mathpzc{y}}(t)\big{)}(y^{\ast})d\mu.$
(6.10)

Proof. It follows from Theorem 6.1 that $\Phi$ is integrably locally
Lipschitzian around $\bar{x}$. Then we apply Theorem 5.2 to observe that the
coderivative Leibniz rule (6.9) holds with the integral of the set-valued
mapping $D^{\ast}\Phi_{t}(\bar{x},\bar{\mathpzc{y}}(t))$ on the the right-hand
side. Applying now the coderivative chain rule for deterministic
multifunctions from (m18, , Theorem 3.11(iii)) under the amenability
assumption of Definition 4, we arrive at the claimed assertion (6.9). Finally,
the limiting coderivative inclusion (6.10) is verified similarly to the above
proof by involving the corresponding arguments of Theorem 5.2. $\hfill\square$

The concluding part of this section provides an efficient specification of the
main Theorem 6.2 for the case of random inequality constraint systems
described by smooth functions. In the deterministic framework, such constraint
systems appear in nonlinear programming, while in the random setting under
consideration they address parametric sets of feasible solutions in problems
of stochastic programming. Let us formalize this as follows. Given
$\varphi^{i}\colon T\times\mathbb{R}^{q}\times\mathbb{R}^{m}\to\mathbb{R}$ a
finite family of convex normal integrands with $i\in I$, assume that
$\varphi^{i}_{t}(z,y)$ are continuously differentiable with respect to $(z,y)$
around the reference points for a.e. $t\in T$ and then define the set-valued
convex normal integrand $F\colon T\times\mathbb{R}^{q}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ by

$F(t,z):=\big{\\{}y\in\mathbb{R}^{m}\;\big{|}\;\varphi^{i}_{t}(z,y)\leq
0\;\text{ for all }\;i\in I\big{\\}}.$ (6.11)

For brevity, we skip here establishing specifications of Theorem 6.1 on
Lipschitz stability for the case of integrands (6.11), while concentrating on
deriving coderivative Leibniz rule for such systems. To proceed, let us
introduce two constraint qualification conditions. The first qualification
condition can be treated as a random uniform version of the classical Slater
constraint qualification in deterministic convex programming. The second one
requires the solution triviality for a certain adjoint system generated by the
derivatives of $\varphi^{i}_{t}$ and $g_{t}$ at he reference points.

######  Definition 5 (integrable constraint qualifications)

Consider the random constraint system (6.11).
(i) Given a measurable mapping $\mathpzc{z}\colon T\to\mathbb{R}^{q}$, we say
that the constraint system (6.11) satisfies the integrable Slater constraint
qualification at $\mathpzc{z}$ if there is a number $\eta>0$ such that for all
$h\in\eta\mathbb{B}$ there exists
$\mathpzc{y}\in\textnormal{L}^{1}(T,\mathbb{R}^{m})$ ensuring the strict
inequality

$\varphi^{i}_{t}\big{(}\mathpzc{z}(t)+h,\mathpzc{y}(t)\big{)}<0\;\text{
whenever }\;i\in I\;\text{ for a.e. }\;t\in T.$ (6.12)

(ii) Fix $x\in{\mathbb{R}^{n}}$, $t\in T$, and $y\in F_{t}(g_{t}(x))$, and
then define the adjoint system at $(t,x,y)$ by

$(z^{\ast},0)=\sum_{i\in
I_{t}(x,y)}\lambda_{i}\nabla\varphi^{i}_{t}\big{(}g_{t}(x),y\big{)},\;\lambda_{i}\geq
0,\;\nabla g_{t}(x)^{\ast}(z^{\ast})=0,$ (6.13)

where $I_{t}(x,y):=\\{i\in I\;|\;\varphi^{i}_{t}(g(x),y)=0\\}$. We say that
the integral triviality qualification condition (ITQC) holds at $\bar{x}$ if
there exists $\eta>0$ such that for all $x\in\mathbb{B}_{\eta}(\bar{x})$, a.e.
$t\in T$, and all $y\in F_{t}(g_{t}(x))$ the adjoint system (6.13) admits only
the trivial solution $z^{\ast}=0$.

The following consequence of Theorem 6.2 establishes pointwise coderivative
Leibniz rules for expected-integral multifunctions $\mathrm{E}_{\Phi}$ with
composite integrands (6.1), where $F$ is represented in the constraint form
(6.11). The obtained coderivative estimates are given entirely in terms of the
initial constraint data.

######  Corollary 2 (coderivative Leibniz rules over random constraint
systems)

Let $\Phi\colon T\times{\mathbb{R}^{n}}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ be an integrably bounded normal
integrand on a complete finite measure space $(T,{\cal A},\mu)$ given in the
composite form (6.1) around $\bar{x}$ with some fixed pair
$(\bar{x},\bar{y})\in\operatorname{gph}\mathrm{E}_{\Phi}$, where $F\colon
T\times\mathbb{R}^{q}\;{\lower
1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise
2.0pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m}$ is the constraint convex-graph
multifunction defined in (6.11), and where $g\colon T\times
U\to\mathbb{R}^{q}$ is measurable in $t$ and continuously differentiable in
$x$ while satisfying

$\sup\big{\\{}\|\nabla g_{t}(x)\|\;\big{|}\;(t,x)\in U\times
T\big{\\}}<\infty.$ (6.14)

Assume further that the ITQC holds at $\bar{x}$, and that the integrable
Slater constraint qualification is satisfied at $\mathpzc{z}:=g_{t}(\bar{x})$.
Then for every $\bar{\mathpzc{y}}\in\mathcal{S}_{\Phi}(\bar{x},\bar{y})$ and
every $y^{\ast}\in\mathbb{R}^{m}$ we have the inclusion

$\widehat{D}^{\ast}\mathrm{E}_{\Phi}(\bar{x},\bar{y})(y^{\ast})\subset\int_{T}\nabla
g_{t}(\bar{x})^{\ast}\circ
D^{\ast}F_{t}\big{(}g_{t}(\bar{x}),\bar{\mathpzc{y}}(t)\big{)}(y^{\ast})d\mu,$
(6.15)

where the limiting coderivative
$D^{\ast}F_{t}(g_{t}(\bar{x}),\bar{\mathpzc{y}}(t))$ is computed by

$\displaystyle
D^{\ast}F_{t}\big{(}g_{t}(\bar{x}),\bar{\mathpzc{y}}(t)\big{)}(y^{\ast})=\left\\{z^{\ast}\in\mathbb{R}^{q}\;\Bigg{|}\;\begin{array}[]{c}(z^{\ast},-y^{\ast})=\displaystyle\sum\limits_{i\in
I_{t}(\bar{x},\bar{y}(t))}\lambda_{i}\nabla\varphi^{i}_{t}\big{(}g_{t}(\bar{x}),\bar{y}(t)\big{)}\\\
\vspace{-0.3cm}\hfil\\\ \text{ for some }\;\lambda_{i}\geq 0\;\text{ with
}\;i\in I_{t}\big{(}\bar{x},\bar{y}(t)\big{)}\end{array}\right\\}.$ (6.19)

If in addition the multifunction $\mathcal{S}_{\Phi}$ is inner semicompact at
$(\bar{x},\bar{y})$, then we have

$D^{\ast}\mathrm{E}_{\Phi}(\bar{x},\bar{y})(y^{\ast})\subset\bigcup\limits_{\bar{\mathpzc{y}}\in\mathcal{S}_{\Phi}(\bar{x},\bar{y})}\int_{T}\nabla
g_{t}(\bar{x})^{\ast}\circ{D}^{\ast}F_{t}\big{(}\bar{x},\bar{\mathpzc{y}}(t)\big{)}(y^{\ast})d\mu,$
(6.20)

where $D^{\ast}F_{t}\big{(}\bar{x},\bar{\mathpzc{y}}(t))$ is computed in
(6.19).

Proof. First we check that all the assumptions of Theorem 6.2 are fulfilled
under the assumptions made in this corollary. Indeed, the required (local)
uniform boundedness of the gradients $\nabla g_{t}(x)$ is formalized in
(6.14), while the imposed integrable Slater constraint qualification (6.12)
readily ensures the existence of $\eta_{1}>0$ for which condition (6.6) holds
whenever $h\in\mathbb{B}_{\eta_{1}}(\bar{x})$. Further, choose $\eta_{2}>0$
such that for all $x\in\mathbb{B}_{\eta_{2}}(\bar{x})$, a.e. $t\in T$, and all
$y\in F_{t}(g_{t}(x))$ we have that the adjoint system (6.13) admits only the
trivial solution $z^{\ast}=0$. Denoting
$\eta:=\min\\{\eta_{1}/(\ell+1),\eta_{2}\\}$ and fixing the triple $(x,y,t)$
from the above ensure that $g_{t}(x)-g_{t}(\bar{x})\in\eta_{1}\mathbb{B}$.
Employing again the integrable Slater condition together with (m06, ,
Corollary 4.35) (by taking into account that the Slater condition yields in
this setting the Mangasarian-Fromovitz constraint qualification assumed in
(m06, , Corollary 4.35(b))), we arrive at the coderivative calculation

$\displaystyle
D^{\ast}F_{t}\big{(}g_{t}(x),y\big{)}(v^{\ast})=\left\\{z^{\ast}\in\mathbb{R}^{q}\;\Bigg{|}\;\begin{array}[]{c}(z^{\ast},-v^{\ast})=\displaystyle\sum\limits_{i\in
I_{t}(x,y)}\lambda_{i}\nabla\varphi^{i}_{t}\big{(}g_{t}(x),y\big{)}\\\
\vspace{-0.3cm}\hfil\\\ \text{ for some }\;\lambda_{i}\geq 0\;\text{ with
}\;i\in I_{t}(x,y)\end{array}\right\\}$

whenever $v^{*}\in\mathbb{R}^{m}$; this clearly implies (6.19). It now follows
from the ITQC that the qualification condition (6.8) is satisfied, and thus
$\Phi$ is an integrably amenable multifunction at $(\bar{x},\bar{y})$.