\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 154, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/154\hfil Absolutely continuous functions]
{Absolutely continuous functions of two variables in the sense of
Carath\'eodory}

\author[J. \v{S}remr\hfil EJDE-2010/154\hfilneg]
{Ji\v{r}\'{\i} \v{S}remr}

\address{Ji\v{r}\'{\i} \v{S}remr \newline
Institute of Mathematics\\
Academy of Sciences of the Czech epublic\\
\v{Z}i\v{z}kova 22\\
616 62 Brno, Czech Republic}
\email{sremr@ipm.cz}

\thanks{Submitted March 19, 2009. Published October 28, 2010.}
\thanks{Supported by grants 201/06/0254 from the Czech Science
Foundation, and AV0Z10190503 \hfill\break\indent
from the Academy of Sciences of the Czech Republic}

\subjclass[2000]{26B30, 26B05}
\keywords{Absolutely continuous function; Carath\'eodory sense;
\hfill\break\indent
integral representation; derivative of double integral}

\begin{abstract}
 In this note, the notion of absolute continuity of functions of two
 variables is discussed. We recall that the set of functions of two
 variables absolutely continuous in the sense of Carath\'eodory
 coincides with the class of functions admitting a certain integral
 representation. We show that absolutely continuous functions in the
 sense of Carath\'eodory can be equivalently characterized in terms
 of their properties with respect to each of variables.  These
 equivalent characterizations play an important role in the
 investigation of boundary value problems for partial differential
 equation of hyperbolic type with discontinuous right-hand side. We
 present several statements which are rather important when analyzing
 strong solutions of such problems by using the methods of real
 analysis but, unfortunately, are not formulated and proven precisely
 in the existing literature, which mostly deals with weak solutions
 or the case where the right-hand side of the equation is continuous.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}

The notion of absolute continuity of functions of a single
variable has been generalized to more variables in many ways. Each
of these constructions observes only some of properties known in
one dimensional case like continuity, differentiability almost
everywhere or integration by parts, the others are naturally lost.
Let us mention absolute continuity due to Schwartz (see
\cite{b36}), absolutely continuous functions in the sense of
Banach or in the sense of Tonelli (see, e.\,g., \cite{b34,b35}),
and $n$-absolute continuity introduced by Mal\'{y} (see
\cite{a239}). We are interested in the notion of absolute
continuity of functions of two variables in the sense of
Carath\'eodory (see \cite{b29}) in order to define a solution of
a partial differential equation of hyperbolic type with
discontinuous right-hand side. More precisely, if we consider the
hyperbolic equation
\begin{equation}
\label{eq:1}
\frac{\partial u(t,x)}{\partial t\,\partial x}=f(t,x,u,u_t,u_x)
\end{equation}
on the rectangle $[a,b]\times[c,d]$, where
$f\colon[a,b]\times[c,d]\times\mathbb{R}^3\to\mathbb{R}$ is
a Carath\'eodory function, then by a solution of the equation
\eqref{eq:1} is usually understood a function $u$ absolutely
continuous on $[a,b]\times[c,d]$ in the sense of Carath\'eodory,
which satisfies the equality \eqref{eq:1} almost everywhere on the
rectangle $[a,b]\times[c,d]$ (see, e.\,g., \cite{a536,a230,a232,a233}).

The aim of this note is to recall the definition of an~absolutely
continuous function of two variables introduced by Carath\'eodory in
1918 and to show that the set of those functions coincides with the
class of functions admitting a certain integral
representation. Some basic properties of such functions like
the existence of partial derivatives and differentiability almost
everywhere are studied, e.\,g., in \cite{b29,b37,a236,a240} (see also
survey given in \cite{a231}). However, for the investigation of the
initial and boundary value problems for the equation \eqref{eq:1}, we
need to show that absolutely continuous functions in the
sense of Carath\'eodory can be \textit{equivalently} characterized
in terms of their properties with respect to each of variables.

\section{Notation and definitions}
\label{sec:notation-definitions}

Throughout this paper, $\mathcal{D}=[a,b]\times[c,d]$ denotes the rectangle
in $\mathbb{R}^2$ and
\begin{equation} \label{eq:8}
Q(t,x)=[a,t]\times[c,x]\quad (t,x) \in\mathcal{D}.
\end{equation}
As usual, $L\big(\mathcal{D};\mathbb{R}\big)$ stands for the set of Lebesgue
integrable functions on $\mathcal{D}$. Further,
$AC\big([\alpha,\beta];\mathbb{R}\big)$ and
$L\big([\alpha,\beta];\mathbb{R}\big)$ are the sets of absolutely
continuous and Lebesgue integrable, respectively, functions on
$[\alpha,\beta]\subset\mathbb{R}$. For any measurable set
$E\subset\mathbb{R}^n$ ($n=1,2$), $\operatorname{meas} E$ denotes the Lebesgue
measure of $E$.

We first introduce a definition of an absolutely continuous function
of two variables (Definition~\ref{def:3}), which is based on the
definition given by Carath\'eodory in 1918. Although one of the
conditions appearing in Definition~\ref{def:3} is slightly different
from those stated in Carath\'eodory's monograph \cite{b29}, both
definitions describe one and the same set of functions (see
Remarks~\ref{rem:1} and \ref{rem:4}).

Let $\mathfrak{S}(\mathcal{D})$ denote the system of all rectangles
$[t_1,t_2]\times[x_1,x_2]$ contained in $\mathcal{D}$. For any rectangle
$P\in\mathfrak{S}(\mathcal{D})$, $|P|$ denotes the \textit{volume} of $P$. We say that
rectangles $P_1,P_2\in\mathfrak{S}(\mathcal{D})$ \textit{do not overlap} if the have no
interior points in common. Furthermore, rectangles $P_1,P_2\in\mathfrak{S}(\mathcal{D})$ are
referred to be \textit{adjoining} if they do not overlap and
$P_1\cup P_2\in\mathfrak{S}(\mathcal{D})$.

\begin{definition}[{\cite[\S 5.3]{b23}}]
\label{def:1} \rm
A finite function $F\colon\mathfrak{S}(\mathcal{D})\to\mathbb{R}$ is said to be
\textit{additive function of rectangles} if, for any adjoining
rectangles $P_1,P_2\in\mathfrak{S}(\mathcal{D})$, the relation
$$
F(P_1\cup P_2)=F(P_1)+F(P_2)
$$
holds.
\end{definition}

\begin{definition}[{\cite[\S 7.3]{b23}}]
\label{def:2} \rm
An additive function of rectangles
$F\colon\mathfrak{S}(\mathcal{D})\to\mathbb{R}$ is \textit{absolutely continuous} if, for
every $\varepsilon>0$, there exists $\delta>0$ such that if
$P_1,\dots,P_k\in\mathfrak{S}(\mathcal{D})$ are mutually non-overlapping rectangles with
the property
$$
\sum_{j=1}^k|P_j|\leq\delta
$$
implies
$$
\sum_{j=1}^k\big|F(P_j)\big|\leq\varepsilon\,.
$$
\end{definition}

The following statement gives a sufficient and necessary condition for
the function of rectangles to be absolutely continuous.

\begin{theorem}[{\cite[\S 7.3]{b23}}]
\label{thm:1}
The function of rectangles $F\colon\mathfrak{S}(\mathcal{D})\to\mathbb{R}$ is absolutely
continuous if and only if there exists a function $h\in L\big(\mathcal{D};\mathbb{R}\big)$ such that
\begin{equation}
\label{eq:2}
F(P)=\iint_{P}h(t,x)\,\mathrm{d}{t}\,\mathrm{d}{x}\quad\text{for }P\in\mathfrak{S}(\mathcal{D}).
\end{equation}
\end{theorem}

Now we are in position to define an absolutely continuous function
of two variables in the sense of Carath\'eodory. Having the function
$u\colon\mathcal{D}\to\mathbb{R}$, we put
\begin{equation}\label{eq:3}
F_u\big([t_1,t_2]\times[x_1,x_2]\big)=u(t_1,x_1)-u(t_1,x_2)-
u(t_2,x_1)+u(t_2,x_2)
\end{equation}
for $[t_1,t_2]\times[x_1,x_2]\in\mathfrak{S}(\mathcal{D})$.
The function of rectangles $F_u\colon\mathfrak{S}(\mathcal{D})\to\mathbb{R}$ defined by the
formula \eqref{eq:3} is said to be \textit{a function of rectangles
associated with $u$}.

\begin{definition} \label{def:3} \rm
We say that a function $u\colon\mathcal{D}\to\mathbb{R}$ is
\textit{absolutely continuous on $\mathcal{D}$ in the sense of
Carath\'eodory} if the following two conditions hold:
\begin{itemize}
\item[(a)]
the function of rectangles $F_u$ associated with $u$ is absolutely
continuous;

\item[(b)]
the functions $u(a,\cdot)\colon[a,b]\to\mathbb{R}$ and
$u(\cdot,c)\colon[c,d]\to\mathbb{R}$ are absolutely continuous.
\end{itemize}
\end{definition}

\begin{remark} \label{rem:1} \rm
In Carath\'eodory's  monograph \cite{b29}, the condition (a)
of the previous definition is slightly different. The construction of
Carath\'eodory reads, roughly speaking, as follows.

Let $\mathfrak{T}(\mathcal{D})$ denote the system of all sets $\sigma$ of
the form
$$
\sigma=\delta_1\cup\dots\cup\delta_p\,,
$$
where $\delta_1,\dots,\delta_p$ is a finite system of mutually
disjoint \textit{open} rectangles contained in $\mathcal{D}$,
and let $\mathfrak{C}$ be the system of all measurable subset of
$\mathcal{D}$. An interval function $\Phi\colon\mathfrak{T}(\mathcal{D})\to\mathbb{R}$ is said to
be \textit{additive and absolutely continuous} if there exists
a set function $G\colon\mathfrak{C}\to\mathbb{R}$ that is
additive, absolutely continuous, and such that
$$
\Phi(\sigma)=G(\sigma)\quad\text{for }\sigma\in\mathfrak{T}(\mathcal{D}).
$$
\end{remark}

Having the function $u\colon\mathcal{D}\to\mathbb{R}$,
according to Carath\'eodory, we put
\begin{equation}
\label{eq:25}
\Phi\big(]t_1,t_2[\times]x_1,x_2[\big)=
u(t_1,x_1)-u(t_1,x_2)-u(t_2,x_1)+u(t_2,x_2)
\end{equation}
for any $]t_1,t_2[\times]x_1,x_2[\subseteq\mathcal{D}$, and
\begin{equation}
\label{eq:13}
\Phi(\sigma)=\Phi(\delta_1)+\cdots+\Phi(\delta_p)\quad\text{for }
\sigma=\delta_1\cup\dots\cup\delta_p\in\mathfrak{T}(\mathcal{D}).
\end{equation}
The interval function $\Phi\colon\mathfrak{T}(\mathcal{D})\to\mathbb{R}$ defined by the
formulae \eqref{eq:25} and \eqref{eq:13} is called \textit{an interval
function associated with $u$}.

\begin{definition}[Carath\'eodory, {\cite[\S567]{b29}}]
\label{def:4}\rm
A function $u\colon\mathcal{D}\to\mathbb{R}$ is said to be
\textit{absolutely continuous} if the following two conditions
hold:
\begin{itemize}
\item[(a)]
the interval function $\Phi$ associated with $u$ is absolutely
continuous;

\item[(b)]
the functions $u(a,\cdot)\colon[a,b]\to\mathbb{R}$ and
$u(\cdot,c)\colon[c,d]\to\mathbb{R}$ are absolutely continuous.
\end{itemize}
\end{definition}

\begin{remark}
\label{rem:4} \rm
Carath\'eodory also proved (see \cite[\S 453, Satz~1]{b29})
that for an interval function $\Phi\colon\mathfrak{T}(\mathcal{D})\to\mathbb{R}$ to be
additive and absolutely continuous it is sufficient and necessary
the existence of a function $p\in L\big(\mathcal{D};\mathbb{R}\big)$ such that
$$
\Phi(\sigma)=\iint_{\sigma}p(t,x)\,\mathrm{d}{t}\,\mathrm{d}{x}\quad\text{for }\sigma\in\mathfrak{T}(\mathcal{D}).
$$
Consequently, by virtue of Theorem~\ref{thm:1}, \textit{the function
$u\colon\mathcal{D}\to\mathbb{R}$ is absolutely continuous in the sense of
Definition~\ref{def:3} if and only if it is absolutely continuous in
the sence of Definition~\ref{def:4}.}
\end{remark}

\section{Main result} \label{sec:main-result}

The main result of this note is the next theorem which is partly
contained in \cite{b29,a236,a240}.

\begin{theorem} \label{thm:2}
The following three statements are equivalent:
\begin{enumerate}
\item
\label{item:1}
the function $u\colon\mathcal{D}\to\mathbb{R}$ is absolutely continuous on
$\mathcal{D}$ in the sense of Carath\'eodory, i.\,e.,
\begin{enumerate}
\renewcommand{\theenumii}{$\alpha$}
\item
\label{item:4}
the function of rectangles $F_u$ associated with $u$ is absolutely
continuous;
\renewcommand{\theenumii}{$\beta$}
\item
\label{item:5}
$u(a,\cdot)\in AC\big([c,d];\mathbb{R}\big)$ and $u(\cdot,c)\in AC\big([a,b];\mathbb{R}\big)$;
\end{enumerate}

\item
\label{item:2}
the function $u\colon\mathcal{D}\to\mathbb{R}$ admits the integral
representation
\begin{equation}
\label{eq:4}
u(t,x)=e+\int_a^tf(s)\,\mathrm{d}{s}+\int_c^xg(\eta)\,\mathrm{d}{\eta}+
\iint_{Q(t,x)}h(s,\eta)\,\mathrm{d}{s}\,\mathrm{d}{\eta}
\end{equation}
for $(t,x)\in\mathcal{D}$,
where $e\in\mathbb{R}$, $f\in L\big([a,b];\mathbb{R}\big)$, $g\in L\big([c,d];\mathbb{R}\big)$, 
$h\in L\big(\mathcal{D};\mathbb{R}\big)$, and the
mapping $Q$ is defined by the formula \eqref{eq:8};

\item
\label{item:3}
the function $u\colon\mathcal{D}\to\mathbb{R}$ satisfies the following
conditions:
\begin{enumerate}
\renewcommand{\theenumii}{\alph{enumii}}
\item%[(a)]
\label{item:6}
$u(\cdot,x)\in AC\big([a,b];\mathbb{R}\big)$ for every $x\in[c,d]$,
$u(a,\cdot)\in AC\big([c,d];\mathbb{R}\big)$;

\item%[(b)]
\label{item:7}
$u_t(t,\cdot)\in AC\big([c,d];\mathbb{R}\big)$ for almost every $t\in[a,b]$;

\item%[(c)]
\label{item:8}
$u_{tx}\in L\big(\mathcal{D};\mathbb{R}\big)$;
\end{enumerate}
\end{enumerate}
\end{theorem}

\begin{remark}
\label{rem:2} \rm
It is clear that, using the Fubini theorem
(see, e.\,g., \cite[XII, \S4, Thm.~1]{b31}), we get
\begin{equation}
\label{eq:28}
\iint_{Q(t,x)}h(s,\eta)\,\mathrm{d}{s}\,\mathrm{d}{\eta}=
\int_a^t\int_c^xh(s,\eta)\,\mathrm{d}{\eta}\,\mathrm{d}{s}=
\int_c^x\int_a^th(s,\eta)\,\mathrm{d}{s}\,\mathrm{d}{\eta}
\quad (t,x) \in\mathcal{D},
\end{equation}
and thus the conditions \eqref{item:6}--\eqref{item:8} in
Theorem~\ref{thm:2} can be replaced by the symmetric ones:
\begin{enumerate}
\renewcommand{\theenumi}{\Alph{enumi}}
\item%[(A)]
\label{item:9}
$u(t,\cdot)\in AC\big([c,d];\mathbb{R}\big)$ for every $t\in[a,b]$,
$u(\cdot,c)\in AC\big([a,b];\mathbb{R}\big)$;

\item%[(B)]
\label{item:10}
$u_x(\cdot,x)\in AC\big([a,b];\mathbb{R}\big)$ for almost every $x\in[c,d]$;

\item%[(C)]
\label{item:11}
$u_{xt}\in L\big(\mathcal{D};\mathbb{R}\big)$.
\end{enumerate}
\end{remark}

\begin{remark}
\label{rem:5} \rm
It follows from Proposition~\ref{prop:1} below that an arbitrary
function $u$ absolutely continuous on $\mathcal{D}$ in the sense of
Carath\'eodory has both mixed second-order derivatives
almost everywhere on the rectangle $\mathcal{D}$ and these derivatives are
equivalent, i.\,e.,
$$
u_{tx}(t,x)=u_{xt}(t,x)\quad\text{for a. e. }(t,x)\in\mathcal{D}.
$$
Moreover, by virtue of Lemma \ref{lem:2} below, one can see that $u$
has also integrable on $\mathcal{D}$ both first-order partial derivatives.
\end{remark}

\begin{remark}
\label{rem:3} \rm
It immediately follows from the definition of an absolutely continuous
function of a single variable that if $v\in AC\big([a,b];\mathbb{R}\big)$ then also
$|v|\in AC\big([a,b];\mathbb{R}\big)$. Unfortunately, functions of two variables \textit{do
not have} this property. Indeed, the function $u$ defined by the
formula
$$
u(t,x)=t-x\quad (t,x) \in[0,1]\times[0,1]
$$
is absolutely continuous on $[0,1]\times[0,1]$ in the sense of
Carath\'eodory but $|u|$ is not, because the condition
\eqref{item:7} of Theorem~\ref{thm:2} does not hold for $|u|$.
\end{remark}

To prove Theorem~\ref{thm:2} we need the following proposition.

\begin{proposition}
\label{prop:1}
Let $h\in L\big(\mathcal{D};\mathbb{R}\big)$ and
\begin{equation}
\label{eq:5}
u(t,x)=
\iint_{Q(t,x)}h(s,\eta)\,\mathrm{d}{s}\,\mathrm{d}{\eta}\quad\text{for }
(t,x)\in\mathcal{D},
\end{equation}
where the mapping $Q$ is defined by the formula \eqref{eq:8}. Then:
\begin{enumerate}
\renewcommand{\theenumi}{\roman{enumi}}
\item%[(i)]
\label{item:12}
there exists a set $E_1\subseteq[a,b]$ such that $\operatorname{meas} E_1=b-a$ and
\begin{equation}
\label{eq:6}
u_t(t,x)=\int_c^xh(t,\eta)\,\mathrm{d}{\eta}\quad\text{for }t\in E_1\ \text{and}\
x\in[c,d];
\end{equation}

\item%[(ii)]
\label{item:13}
there exists a set $E_2\subseteq[c,d]$ such that $\operatorname{meas} E_2=d-c$ and
$$
u_x(t,x)=\int_a^th(s,x)\,\mathrm{d}{s}\quad\text{for }t\in[a,b]\ \text{and}\
x\in E_2;
$$

\item%[(iii)]
\label{item:14}
there exists a set $E_3\subseteq\mathcal{D}$ such that
$\operatorname{meas} E_3=(b-a)(d-c)$ and
\begin{equation}
\label{eq:7}
u_{tx}(t,x)=h(t,x)\quad\text{for }(t,x)\in E_3;
\end{equation}

\item%[(iv)]
\label{item:15}
there exists a set $E_4\subseteq\mathcal{D}$ such that
$\operatorname{meas} E_4=(b-a)(d-c)$ and
$$
u_{xt}(t,x)=h(t,x)\quad\text{for }(t,x)\in E_4.
$$
\end{enumerate}
\end{proposition}

Parts \eqref{item:12}, \eqref{item:13} are proved in \cite[\S7]{a236},
the proofs of parts \eqref{item:14}, \eqref{item:15} are based on the
proof of Lemma 1 stated in \cite{a240}. For the sake of completeness
we prove Proposition~\ref{prop:1} in detail.

\begin{lemma}
\label{lem:2}
Let $h\in L\big(\mathcal{D};\mathbb{R}\big)$ and
\begin{equation}
\label{eq:17}
\varphi(t,x)=\int_c^xh(t,\eta)\,\mathrm{d}{\eta}\quad\text{for }t\in E,\
x\in[c,d],
\end{equation}
where $E\subseteq[a,b]$ is such that $\operatorname{meas} E=b-a$. Then the function
$\varphi$ is measurable on the rectangle $\mathcal{D}$.
\end{lemma}

This lemma is stated in \cite[\S569]{b29}; we give here a slightly
simpler proof.

\begin{proof}%{Proof of Lemma \ref{lem:2}.}
It is clear that $E\times[c,d]$ is measurable subset of the
rectangle $\mathcal{D}$ with the measure equal to $(b-a)(d-c)$, and thus the
function $\varphi$ is defined almost everywhere on $\mathcal{D}$.

First suppose that the function $h$ is bounded, i.\,e.,
$$
|h(t,x)|\leq M\quad\text{for a. e. }(t,x)\in\mathcal{D}.
$$
It is well-known (see, e.\,g., \cite[XII, \S1]{b31}) that there
exists a sequence $\{f_n\}_{n=1}^{+\infty}$ of functions continuous on
$\mathcal{D}$ such that
\begin{equation}
\label{eq:18}
\lim_{n\to+\infty}f_n(t,x)=h(t,x)\quad\text{for a. e. }(t,x)\in\mathcal{D}.
\end{equation}
Without loss of generality we can assume that
\begin{equation}
\label{eq:19}
|f_n(t,x)|\leq M\quad (t,x) \in\mathcal{D},\; n\in\mathbb{N}.
\end{equation}
According to \eqref{eq:18} and the Fubini theorem
(see, e.\,g., \cite[XI, \S5, Cor.~1]{b31}), there exists a set
$F\subseteq[a,b]$ such that $\operatorname{meas} F=b-a$ and the
condition
\begin{equation}
\label{eq:20}
\lim_{n\to+\infty}f_n(t_0,x)=h(t_0,x)\quad \text{for a. e.}  x\in[c,d].
\end{equation}
holds  for every $t_0\in F$. Now we put
\begin{equation}
\label{eq:21}
\psi_n(t,x)=\int_c^xf_n(t,\eta)\,\mathrm{d}{\eta}\quad (t,x) \in\mathcal{D},\
n\in\mathbb{N}.
\end{equation}
It is easy to verify that the functions $\psi_n$ are continuous on
$\mathcal{D}$. Indeed, by virtue of \eqref{eq:19}, for any
$(t_1,x_1),(t_2,x_2)\in\mathcal{D}$ and $n\in\mathbb{N}$ we get
\begin{align*}
|\psi_n(t_2,x_2)-\psi_n(t_1,x_1)|
&= \big|\int_c^{x_2}f_n(t_2,\eta)\,\mathrm{d}{\eta}-
\int_c^{x_1}f_n(t_1,\eta)\,\mathrm{d}{\eta}\big|\\
&\leq\int_c^d|f_n(t_2,\eta)-f_n(t_1,\eta)|\,\mathrm{d}{\eta}+
M|x_2-x_1|,
\end{align*}
and thus continuity of $\psi_n$ on $\mathcal{D}$ follows from continuity
of $f_n$. Consequently, the functions $\psi_n$ are measurable on
$\mathcal{D}$.

On the other hand, in view of \eqref{eq:19}--\eqref{eq:21},
the Lebesgue convergence theorem yields
\[
\varphi(t,x)=\int_c^xh(t,\eta)\,\mathrm{d}{\eta}
=\lim_{n\to+\infty}\int_c^xf_n(t,\eta)\,\mathrm{d}{\eta}
=\lim_{n\to+\infty}\psi_n(t,x)
\]
for $t\in E\cap F,\; x\in[c,d]$;
i.\,e.,
$$
\varphi(t,x)=\lim_{n\to+\infty}\psi_n(t,x)\quad\text{for a. e. }(t,x)\in\mathcal{D}.
$$
Therefore, the function $\varphi$ is measurable on $\mathcal{D}$
(see, e.\,g., \cite[XII, \S1]{b31}).

Now suppose that the function $h$ is unbounded. For any
$k\in\mathbb{N}$, we put
$$
h_k(t,x)=
\begin{cases}
-k &\text{for } (t,x)\in\mathcal{D},\; h(t,x)<-k,\\
h(t,x) &\text{for } (t,x)\in\mathcal{D},\; |h(t,x)|\leq k,\\
k &\text{for } (t,x)\in\mathcal{D},\; h(t,x)>k.
\end{cases}
$$
It is clear that the functions $h_k$ $(k\in\mathbb{N})$ are measurable
and bounded on $\mathcal{D}$ and, for any $t_0\in E$, we have
\begin{equation}
\label{eq:30}
|h_k(t_0,x)|\leq|h(t_0,x)|\quad \text{for a. e.}  x\in[c,d],\ k\in\mathbb{N}
\end{equation}
and
\begin{equation}
\label{eq:31}
h(t_0,x)=\lim_{k\to+\infty}h_k(t_0,x)\quad \text{for a. e.}
 x\in[c,d].
\end{equation}
Moreover, according to the above-proved, the functions $\varphi_k$
$(k\in\mathbb{N})$ defined by
$$
\varphi_k(t,x)=\int_c^xh_k(t,\eta)\,\mathrm{d}{\eta}\quad\text{for }t\in E,\;
x\in[c,d],\; k\in\mathbb{N}
$$
are measurable on $\mathcal{D}$.  On the other hand, in view of
\eqref{eq:30} and \eqref{eq:31}, it follows from the
Lebesgue convergence theorem that
$$
\lim_{k\to+\infty}\int_c^xh_k(t,\eta)\,\mathrm{d}{\eta}=
\int_c^xh(t,\eta)\,\mathrm{d}{\eta}\quad\text{for }t\in E,\ x\in[c,d],
$$
i.\,e.,
$$
\varphi(t,x)=\lim_{k\to+\infty}\varphi_k(t,x)\quad\text{for a. e. }(t,x)\in\mathcal{D}.
$$
Consequently, the function $\varphi$ is measurable on $\mathcal{D}$
(see, e.\,g., \cite[XII, \S1]{b31}).
\end{proof}

\begin{proof}[Proof of Proposition~\ref{prop:1}]
We first extent the function $h$ outside of $\mathcal{D}$ by setting
$$
h(t,x)=0\quad (t,x) \in\mathbb{R}^2\setminus\mathcal{D}.
$$

\eqref{item:12} Without loss of generality we can assume that
$$
h(t,x)\geq0\quad\text{for a. e. }(t,x)\in\mathcal{D}
$$
(in contrary case we represent the function $h$ as the difference of
its positive and negative parts).
It follows from \eqref{eq:28} and  \eqref{eq:5} that, for any
$x\in[c,d]$, there exists a set $A(x)\subseteq\,]a,b[$ such that
$\operatorname{meas} A(x)=b-a$ and
\begin{equation}
\label{eq:9}
u_t(t,x)=\int_c^xh(t,\eta)\,\mathrm{d}{\eta}\quad\text{for }x\in[c,d]\ \text{and}\
t\in A(x).
\end{equation}
Put $E_1=\cap_{x\in B}A(x)$, where
$B=\big([c,d]\cap\mathbb{Q}\big)\cup\{c,d\}$. Since the set $B$ is
countable, the set $E_1$ is measurable and $\operatorname{meas} E_1=b-a$. Moreover,
the condition \eqref{eq:9} yields
\begin{equation}
\label{eq:10}
u_t(t,x)=\int_c^xh(t,\eta)\,\mathrm{d}{\eta}\quad\text{for }t\in E_1
 \text{ and } x\in B.
\end{equation}
We will show that the last relation holds for every $x\in[c,d]$.

Let $t\in E_1$ be arbitrary but fixed. Let, moreover,
$\{l_n\}_{n=1}^{+\infty}$ be an arbitrary sequence of nonzero numbers
such that
\begin{equation}
\label{eq:11}
\lim_{n\to+\infty}l_n=0.
\end{equation}
Put
\begin{equation}
\label{eq:12}
\varphi_n(x)=\frac{1}{l_n}\int_t^{t+l_n}\int_c^x
h(s,\eta)\,\mathrm{d}{\eta}\,\mathrm{d}{s}\quad\text{for }x\in[c,d],\ n\in\mathbb{N}.
\end{equation}
According to \eqref{eq:10}--\eqref{eq:12}, we obtain
\begin{equation}
\label{eq:14}
\lim_{n\to+\infty}\varphi_n(x)=
\int_c^xh(t,\eta)\,\mathrm{d}{\eta}\quad\text{for }x\in B.
\end{equation}
Since the function $h$ is assumed to be nonnegative, it is clear that
the functions $\varphi_n$ $(n\in\mathbb{N})$ are nondecreasing on
$[c,d]$.

We will show that the relation \eqref{eq:14} holds for every
$x\in[c,d]$. Let $x_0\in[c,d]$ and $\varepsilon>0$ be arbitrary but
fixed. Then there exist $x_1,x_2\in B$ such that $x_1\leq x_0\leq x_2$
and
\begin{equation}
\label{eq:15}
\int_{x_1}^{x_0}h(t,\eta)\,\mathrm{d}{\eta}<\frac{\varepsilon}{2}\,,\quad
\int_{x_0}^{x_2}h(t,\eta)\,\mathrm{d}{\eta}<\frac{\varepsilon}{2}\,.
\end{equation}
Moreover, by virtue of \eqref{eq:14}, there exists $n_0\in\mathbb{N}$
such that
\begin{equation}
\label{eq:16}
\big|\varphi_n(x_k)-\int_c^{x_k}h(t,\eta)\,\mathrm{d}{\eta}\big|
<\frac{\varepsilon}{2}\quad\text{for }n\geq n_0\,,\; k=1,2.
\end{equation}
Using \eqref{eq:15}, \eqref{eq:16}, and the monotonicity of
$\varphi_n$, we obtain
$$
\varphi_n(x_0)-\int_c^{x_0}h(t,\eta)\,\mathrm{d}{\eta}\leq
\varphi_n(x_2)-\int_c^{x_2}h(t,\eta)\,\mathrm{d}{\eta}+
\int_{x_0}^{x_2}h(t,\eta)\,\mathrm{d}{\eta}<\varepsilon\quad\text{for }
n\geq n_0
$$
and
$$
\int_c^{x_0}h(t,\eta)\,\mathrm{d}{\eta}-\varphi_n(x_0)\leq
\int_{x_1}^{x_0}h(t,\eta)\,\mathrm{d}{\eta}+
\int_c^{x_1}h(t,\eta)\,\mathrm{d}{\eta}-\varphi_n(x_1)<
\varepsilon\quad\text{for }n\geq n_0\,,
$$
and thus
$$
\big|\varphi_n(x_0)-\int_c^{x_0}h(t,\eta)\,\mathrm{d}{\eta}\big|
<\varepsilon\quad\text{for }n\geq n_0.
$$
Consequently, in view of arbitrariness of $x_0$ and $\varepsilon$,
the relation
$$
\lim_{n\to+\infty}\varphi_n(x)=
\int_c^xh(t,\eta)\,\mathrm{d}{\eta}\quad\text{for }x\in[c,d]
$$
holds. Since the sequence $\{l_n\}_{n=1}^{+\infty}$ was also
arbitrary, we have proved that
$$
u_t(t,x)=\int_c^xh(t,\eta)\,\mathrm{d}{\eta}\quad\text{for }x\in[c,d].
$$
Mention on arbitrariness of $t\in E_1$ completes the proof of the part
(i).

\eqref{item:13}
The proof is analogous to the previous case and thus we omit it.

\eqref{item:14}
According to the above-proved part \eqref{item:12}, there exists a set
$E_1\subseteq[a,b]$ such that $\operatorname{meas} E_1=b-a$ and the condition
\eqref{eq:6} holds. Let
\begin{equation}
\label{eq:24}
E_3=\big\{(t,x)\in E_1\times\,]c,d[\,\colon
\lim_{l\to0}\frac{u_t(t,x+l)-u_t(t,x)}{l}=h(t,x)\big\}.
\end{equation}
Then it is clear that, for any $(t,x)\in E_3$, there exists
$u_{tx}(t,x)$ and the condition \eqref{eq:7} is satisfied.
We will show that the set $E_3$ is measurable with the measure equal
to $(b-a)(d-c)$.

Put $W=E_1\times\,]c,d[$ and
\begin{gather*}
\varphi_k(t,x)=\inf_{0<|l|<1/k}\Big\{
\frac{1}{l}\int_x^{x+l}h(t,\eta)\,\mathrm{d}{\eta}\Big\}\quad\text{for }
(t,x)\in W,\ k\in\mathbb{N},\\
\psi_k(t,x)=\sup_{0<|l|<1/k}\Big\{
\frac{1}{l}\int_x^{x+l}h(t,\eta)\,\mathrm{d}{\eta}\Big\}\quad\text{for }
(t,x)\in W,\ k\in\mathbb{N}.
\end{gather*}
For any $(t,x)\in W$ fixed, the function of variable $l$ in braces
is continuous, and thus the functions $\varphi_k$ and $\psi_k$ can
be defined as follows
\begin{gather}
\label{eq:22}
\varphi_k(t,x)=\inf_{0<|l|<1/k,\, l\in\mathbb{Q}}\Big\{
\frac{1}{l}\int_x^{x+l}h(t,\eta)\,\mathrm{d}{\eta}\Big\}\quad\text{for }
(t,x)\in W,\ k\in\mathbb{N},\\
\label{eq:23}
\psi_k(t,x)=\sup_{0<|l|<1/k,\, l\in\mathbb{Q}}\Big\{
\frac{1}{l}\int_x^{x+l}h(t,\eta)\,\mathrm{d}{\eta}\Big\}\quad\text{for }
(t,x)\in W,\ k\in\mathbb{N}.
\end{gather}
It is clear that the set $W$ is measurable with the measure equal to
$(b-a)(d-c)$ and, for any $l\in\mathbb{Q}$ fixed, the function of
$(t,x)$ in braces is measurable on $\mathcal{D}$ (see Lemma \ref{lem:2}),
and thus it is measurable on $W$. Consequently, the functions
$\varphi_k$ and $\psi_k$, which are defined as the point-wise infimum
and supremum of countable system of measurable functions, are
measurable on $W$ (see \cite[XII, \S1]{b31}).   Moreover,
in view of \eqref{eq:22} and \eqref{eq:23}, the relations
$$
\varphi_1(t,x)\leq\varphi_2(t,x)\leq\cdots,\quad
\psi_1(t,x)\geq\psi_2(t,x)\geq\cdots\quad (t,x) \in W
$$
holds. Consequently, there exist point-wise (finite or infinite)
limits
$$
\varphi(t,x)=\lim_{k\to+\infty}\varphi_k(t,x),\quad
\psi(t,x)=\lim_{k\to+\infty}\psi_k(t,x)\quad (t,x) \in W.
$$
Now it follows from measurability of the functions
$\varphi_k$ and $\psi_k$ that the limit functions $\varphi$ and $\psi$
are also measurable. By virtue of \eqref{eq:6}, for any
$(t,x)\in\mathcal{D}$ we obtain
\begin{align*}
\varphi(t,x)
&=\lim_{k\to+\infty} \Big(\inf_{0<|l|<1/k}\Big\{
\frac{1}{l}\int_x^{x+l}h(t,\eta)\,\mathrm{d}{\eta}\Big\}\Big)\\
&=\lim_{\delta\to0+}\Big(\inf_{0<|l|<\delta}\Big\{
\frac{1}{l}\int_x^{x+l}h(t,\eta)\,\mathrm{d}{\eta}\Big\}\Big)\\
&= \liminf_{l\to0}\frac{u_t(t,x+l)-u_t(t,x)}{l}\;.
\end{align*}
One can show in a similar manner that
$$
\psi(t,x)=\limsup_{l\to0}\frac{u_t(t,x+l)-u_t(t,x)}{l}\quad
(t,x) \in W.
$$
Now it is clear that the set $E_3$ defined by the formula
\eqref{eq:24} satisfies
$$
E_3=\{(t,x)\in W\colon\varphi(t,x)=\psi(t,x)=h(t,x)\}.
$$
This set is however measurable, because the function $\varphi$,
$\psi$, and $h$ are measurable on $W$. Moreover, the condition
\eqref{eq:6} guarantees that, for any $t\in E_1$, the section $E_3(t)$
of the set $E_3$ has the measure equal to $b-a$. Therefore,
using the Fubini theorem (see, e.\,g., \cite[XI, \S5, Thm.~1]{b31})
we get
$$
\operatorname{meas} E_3=\int_{E_1}\operatorname{meas} E_3(t)\,\mathrm{d}{t}
=(d-c)\operatorname{meas} E_1=(b-a)(d-c).
$$

\eqref{item:15}
The proof is analogous to the previous case, but the part
\eqref{item:13} has to be used instead of \eqref{item:12}.
\end{proof}

Now we are in position to prove Theorem~\ref{thm:2}.

\begin{proof}[Proof of Theorem~\ref{thm:2}]
To prove the theorem it is sufficient to show that that the
implications $\eqref{item:1}\Rightarrow\eqref{item:2}$,
$\eqref{item:2}\Rightarrow\eqref{item:3}$,
and $\eqref{item:3}\Rightarrow\eqref{item:1}$ are satisfied.

$\eqref{item:1}\Rightarrow\eqref{item:2}$: Suppose that the function
$u\colon\mathcal{D}\to\mathbb{R}$ is absolutely continuous on
$\mathcal{D}$ in the sense of Carath\'eodory, i.\,e., the conditions
\eqref{item:4} and \eqref{item:5} are fulfilled. According to the
condition \eqref{item:5} and Theorem~\ref{thm:1}, there exists a function
$h\in L\big(\mathcal{D};\mathbb{R}\big)$ such that
$$
F_u(P)=\iint_{P}h(t,x)\,\mathrm{d}{t}\,\mathrm{d}{x}\quad\text{for }P\in\mathfrak{S}(\mathcal{D}).
$$
In particular, we have
\begin{equation}
\label{eq:26}
F_u\big(Q(t,x)\big)=\iint_{Q(t,x)}h(s,\eta)
\,\mathrm{d}{s}\,\mathrm{d}{\eta}\quad (t,x) \in\mathcal{D},
\end{equation}
where the mapping $Q$ is defined by the formula \eqref{eq:8}.
Furthermore, the relation \eqref{eq:3} yields
\begin{equation}
\label{eq:27}
F_u\big(Q(t,x)\big)=u(a,c)-u(t,c)-u(a,x)+u(t,x)\quad\text{for }
(t,x)\in\mathcal{D}.
\end{equation}

On the other hand, the condition \eqref{item:4} guarantees that there
exist $f\in L\big([a,b];\mathbb{R}\big)$ and $g\in L\big([c,d];\mathbb{R}\big)$ such that
\begin{gather*}
u(t,c)=u(a,c)+\int_a^tf(s)\,\mathrm{d}{s}\quad\text{for }t\in[a,b],\\
u(a,x)=u(a,c)+\int_c^xg(\eta)\,\mathrm{d}{\eta}\quad\text{for }x\in[c,d].
\end{gather*}
Now, using the last two relations, it follows from \eqref{eq:26}
and \eqref{eq:27} that the function $u$ admits the integral
representation \eqref{eq:4} with $e=u(a,c)$.
\vskip0.5\baselineskip

$\eqref{item:2}\Rightarrow\eqref{item:3}$: Suppose that the function
$u\colon\mathcal{D}\to\mathbb{R}$ admits the integral representation
\eqref{eq:4}, where $e\in\mathbb{R}$, $f\in L\big([a,b];\mathbb{R}\big)$,
$g\in L\big([c,d];\mathbb{R}\big)$, and $h\in L\big(\mathcal{D};\mathbb{R}\big)$. By virtue of the relations \eqref{eq:28},
it is clear that the condition \eqref{item:6} holds.
Moreover, according to
Proposition~\ref{prop:1}\eqref{item:12}, there exists a set
$E_1\subseteq[a,b]$ such  that $\operatorname{meas} E_1=b-a$ and
$$
u_t(t,x)=f(t)+\int_c^xh(t,\eta)\,\mathrm{d}{\eta}\quad\text{for }t\in E_1\
\text{and}\ x\in[c,d].
$$
Therefore, the condition \eqref{item:7} is fulfilled. Finally,
Proposition~\ref{prop:1}\eqref{item:13} guarantees the validity of the
condition \eqref{eq:7}, where $E_3\subseteq\mathcal{D}$ is a measurable set
with the measure equal to $(b-a)(d-c)$. Consequently, the condition
\eqref{item:8} also
holds.

$\eqref{item:3}\Rightarrow\eqref{item:1}$: Suppose that the function
$u\colon\mathcal{D}\to\mathbb{R}$ satisfies the conditions
\eqref{item:6}--\eqref{item:8}. Obviously, the condition
\eqref{item:5} holds.

Let $P=[t_1,t_2]\times[x_1,x_2]\in\mathfrak{S}(\mathcal{D})$ be an arbitrary rectangle.
By virtue of \eqref{eq:3}, we have
$$
F_u(P)=u(t_1,x_1)-u(t_1,x_2)-u(t_2,x_1)+u(t_2,x_2)
$$
and thus, using the conditions \eqref{item:6}--\eqref{item:8}, we obtain
\begin{align*}
F_u(P)&=\int_{t_1}^{t_2}u_s(s,x_2)\,\mathrm{d}{s}-
\int_{t_1}^{t_2}u_s(s,x_1)\,\mathrm{d}{s}\\
&= \int_{t_1}^{t_2}\big[u_s(s,x_2)-u_s(s,x_1)\big]\,\mathrm{d}{s} \\
&=\int_{t_1}^{t_2}\Big(\int_{x_1}^{x_2}u_{s\eta}(s,\eta)
\,\mathrm{d}{\eta}\Big)\,\mathrm{d}{s}
=\iint_{P}u_{s\eta}(s,\eta)\,\mathrm{d}{s}\,\mathrm{d}{\eta}.
\end{align*}
Therefore, in view of the condition \eqref{item:8} and
Theorem~\ref{thm:1}, it is clear that the function of rectangles $F_u$
associated with $u$ is absolutely continuous, i.\,e., the condition
\eqref{item:4} holds. Consequently, the function $u$ is absolutely
continuous on $\mathcal{D}$ in the sense of Carath\'eodory.
\end{proof}

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\end{document}