%
%  MDAI #091
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%     Regularity properties of null-additive fuzzy measure    %
%                     on metric spaces                        %
%                                                             %
%                  J. Li,  M. Yasuda, J. Song                 %
%                                                             %
%    2004/09/16                                               %
%    2005/3/31  lysMDAI091.tex                                %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\documentclass{llncs}
\usepackage{latexsym,amssymb,theorem}
%\renewcommand{\thefootnote}{\fnsymbol{footnote}}
%\newtheorem{prop}{Proposition}[section]
%\newtheorem{df}{Definition}[section]
%\newtheorem{thm}{Theorem}[section]
%\newtheorem{cor}{Corollary}[section]
%\newtheorem{lem}{Lemma}[section]
%\newtheorem{rem}{Remark}[section]
%\newtheorem{exa}{Example}[section]
%
\begin{document}
%\baselineskip=8mm
%-------------------------the text-----------------------
\title{ Regularity properties of null-additive fuzzy measure 
        on metric spaces }
%\titlerunning{null-additive fuzzy measure}

\author{Jun Li\inst{1} 
   \and Masami Yasuda\inst{2} 
   \and Jinjie Song\inst{3}
}

\institute{
  Department of Applied Mathematics, College of Science,
  Communication University of China,
  Beijing 100024, China\thanks{The first author wants to show his thanks to
       the China Scholarship Council.}
  \email{ lijun@seu.edu.cn}
\and 
  Dep of Math \& Infor., Chiba University, Chiba 263-8522 Japan
  \email{yasuda@math.s.chiba-u.ac.jp}
\and 
  Department of Computer Science and Engineering,
  Tianjin University of Technology,
  Tianjin 300191, China
\email{songjinjie@sidewave.com}
%
}
%


%\author{Jun Li}
%{\thanks{The author was supported by
%                        the China Scholarship Council. }} \\
%\small{Dep of Applied Math., Southeast University,} \\
%\small{Nanjing 210096, China } \\[2mm]
%
%        Masami Yasuda{\thanks{
%                       Corresponding author.
%                       E-mail address: yasuda@math.s.chiba-u.ac.jp} } \\
%\small{Dep of Math \& Infor., Chiba University, Chiba 263-8522, Japan}  \\[2mm]
%
%        Jinjie Song \\
%\small{Dep of Technology Research, Sidewave Co., Ltd.} \\
%\small{Shimotsuruma 4374-4-410, Yamato, Kanagawa, 242-0001, Japan}
%}
%
%\date{}
\maketitle
%\frontmatter 
%
\begin{abstract}
We shall discuss further regularity properties of null-additive
fuzzy measure on metric spaces following the previous results.
Under the null-additivity condition, some properties of the
inner/outer regularity and the regularity of fuzzy measure are
shown. Also the strong regularity of fuzzy measure is discussed on
complete separable metric spaces. As an application of strong
regularity, we present a characterization of atom of null-additive
fuzzy measure.

\vspace{2mm} 
{\it Keywords}: Fuzzy measure; null-additivity; regularity;
\end{abstract}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%       Section 1           %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
 \label{Intro}
Recently various regularities of a set function are
proposed and investigated by many authors
(\cite{Jiang96,Li04b,Naru00,Naru03,Naru04,Pap95,Song03,Wu01}). 
As it is seen, the regularities play an important role 
in the nonadditive measure theory. 
In \cite{Li04b,Song03} we discussed the regularity of 
a null-additive fuzzy measure and 
proved Egoroff's theorem and Lusin's theorem for 
fuzzy measures on a metric space.

In this paper, we shall continue to investigate further 
regularities of a fuzzy measure on metric spaces 
following the results by \cite{Li04b,Song03}. 
Explicitly, under the null-additivity, 
the weekly null-additivity and the converse null-additivity 
condition, we shall discuss these relation among 
the inner regularity, the outer regularity and the regularity 
of fuzzy measures. Also we define the strong regularity 
of fuzzy measures and show our main result: 
  the null-additive fuzzy measures possess a strong regularity 
  on complete separable metric spaces. 
By using strong regularity we shall show a version of Egoroff's 
theorem and Lusin's theorem for null-additive fuzzy measures 
on complete separable metric spaces, respectively.
Lastly, as an application of a strong regularity, 
we present a characterization of atom of 
a null-additive fuzzy measure.

In preparation of the paper, authors are told 
two references \cite{NaMu04} and \cite{Pap95b} from anonymous 
referee. We find that there is another characterization of 
atom of null additive set functions in \cite{Pap95b}. 
Also, in \cite{NaMu04}, a similar result of our main result 
is discussed. However, these results are not completely 
consistent with ours.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%       Section 2           
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Preliminaries}
Throughout this paper, we assume that $(X, \ d)$ is a
metric space, and that ${\cal O}$, ${\cal C}$ and ${\cal K}$ are
the classes of all open, closed and compact sets in $(X, d)$,
respectively. ${\cal B}$ denotes Borel $\sigma$-algebra on $X$,
i.e., it is the smallest $\sigma$-algebra containing ${\cal O}$.
Unless stated otherwise all the subsets mentioned are supposed to
belong to ${\cal B}$.

A set function $\mu:{\cal B} \to [0, +\infty]$ is said to be
(i)\,%
{\it continuous from below},
 if $\lim_{n \to \infty}\mu(A_n) = \mu(A)$
whenever $A_n \nearrow A$;
(ii)\,%
 {\it continuous from above}, if
 $\lim_{n \to \infty}\mu(A_n) = \mu(A)$
whenever $A_n \searrow A$;
(iii)\,%
 {\it strongly order continuous}, if
$\lim_{n \to +\infty} \mu(A_n) = 0$ whenever
$A_n \searrow \ B$ and $\mu(B) = 0$;
(iv)\,%
 {\it null-additive}, if
 $\mu(E \cup F)=\mu(E)$ for any $E$ whenever $\mu(F) = 0$;
(v)\,%
 {\it weakly null-additive}, if
 $\mu(E \cup F) = 0$ whenever $ \mu(E) = \mu(F) = 0$;
(vi)\,%
 {\it converse-null-additive}, if
 $\mu(E - F) = 0$ whenever $F \subset E$ and $\mu(F) = \mu(E) <
 +\infty$;
(vii)\,%
  {\it finite}, if $\mu(X) < \infty$.
  
Refer to these definitions and their relations between them 
in \cite{Li03b} etc.
We note here that, obviously, the null-additivity of $\mu$ implies weakly
null-additivity.
%
%%%   Definition 1   %%%
\begin{definition} \
{\rm
 A fuzzy measure on $(X, {\cal B})$ is an extended
real valued set function $\mu:{\cal F} \to [0, +\infty]$
satisfying the following conditions: \\
{\rm (1)} \ \ $\mu(\emptyset)=0$; \\
{\rm (2)} \ \ $\mu(A) \leq \mu(B)$ whenever $A \subset B$ and $A,
B \in {\cal F}$ \ \ {\rm(monotonicity)}.
}
\end{definition}

We say that a fuzzy measure $\mu$ is continuous if it is
continuous both from below and from above.
Our fundamental assumtion in this paper is that  $\mu$ is 
a ``finite'' fuzzy measure.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%       Section 3           
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Regularity of fuzzy measure}
\label{regularity}
%
%%%   Definition 2   %%%
\begin{definition}{\rm (\cite{Wu01})} \
{\rm
 A fuzzy measure $\mu$ is called
   {\it outer regular} (resp. {\it inner regular}),
 if for each $A \in {\cal B}$ and each $\epsilon > 0$,
 there exists a set $G \in {\cal O}$ (resp. $F \in {\cal C}$)
   such that
$A \subset G, \ \mu(G - A) < \epsilon$
(resp.
$F \subset A, \ \mu(A - F) < \epsilon)$.
  $\mu$ is called {\it regular},
 if for each $A \in {\cal B}$ and each $\epsilon > 0$,
  there exist a closed set $F \in {\cal C}$ and an open set $G \in {\cal O}$
   such that $F \subset A \subset G$ and $\mu(G - F) < \epsilon$.
}
\end{definition}
%

Obviously, if fuzzy measure $\mu$ is regular,
then it is both outer regular and inner regular.
%
%%%   Proposition 1   %%%
\begin{proposition}{\rm (\cite{Li04b})}
  \label{Prop-a}
If $\mu$ is weekly null-additive and continuous,
then it is regular. Furthermore, if $\mu$ is null-additive, then
for any $A \in {\cal B}$,
\begin{eqnarray}
\mu(A)
&=& \sup\{\ \mu(F) \ | \ F \subset A, \ F \in {\cal C} \ \} \nonumber \\
&=& \inf\{\ \mu(G) \ | \ G \supset A, \ G \in {\cal O} \ \}
\nonumber
\end{eqnarray}
\end{proposition}

In the following we present some properties of the inner regularity and outer regularity
of fuzzy measure, their proofs can be easily obtained:
%
%%%   Proposition 2   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}
  \label{Prop-b}
If $\mu$ is weekly null-additive and strongly order continuous,
then both outer regularity and inner regularity imply regularity.
\end{proposition}
%
%%%   Proposition 3   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}
  \label{Prop-c}
Let $\mu$ be null-additive fuzzy measure.
\begin{itemize}
  \item[{\rm (1)}] \ If $\mu$ is continuous from below, 
    then inner regularity implies
    \[
      \mu(A) = \sup\{\ \mu(F) \ | \ F \subset A, \ F \in {\cal C}\ \}
    \]
    for all $A \in {\cal B};$
  \item[{\rm (2)}] \ If $\mu$ is continuous from above, then outer
regularity implies
\[
\mu(A) = \inf\{\ \mu(G) \ | \ A \subset G, \ G \in {\cal O} \ \}
\]
for all $A \in {\cal B}$.
\end{itemize}
\end{proposition}

%%%   Proposition 4   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}
  \label{Prop-d}
Let $\mu$ be converse-null-additive fuzzy measure.
\begin{itemize}
  \item[{\rm (1)}] \ If $\mu$ is continuous from below and strongly order
    continuous, and for any $A \in {\cal B}$,
    \[
      \mu(A) = \sup\{\ \mu(F) \ | \ F \subset A, \ F \in {\cal C}\ \},
    \]
    then $\mu$ is inner regular.
  \item[{\rm (2)}] \ If $\mu$ is continuous from above, and for any 
    $A \in {\cal B}$,
    \[
      \mu(A) = \inf\{\ \mu(G) \ | \ A \subset G, \ G \in {\cal O} \ \},
    \]
    then $\mu$ is outer regular.
\end{itemize}
\end{proposition}

%%%   Definition 3   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{definition} \
{\rm
$\mu$ is called {\it strongly regular},
 if for each $A \in {\cal B}$ and each $\epsilon > 0$,
  there exist a compact set $K \in {\cal K}$ and an open set $G \in {\cal O}$
   such that $K \subset A \subset G$ and $\mu(G - K) < \epsilon$.
}
\end{definition}

The strongly regularity implies regularity, and hence inner regularity
and outer regularity.

%%%   Proposition 5   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}
  \label{Prop-e}
Let $\mu$ be null-additive and continuous from below. If
 $\mu$ is strongly regular, then for any $A \in {\cal B}$,
\[
 \mu(A) = \sup\{\ \mu(K) \ | \ K \subset A, \ K \in {\cal K}\ \}.
\]
\end{proposition}

%%%   Proposition 6   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}
  \label{Prop-f}
Let $\mu$ be null-additive and order continuous.
If for any $A \in {\cal B}$,
\[
  \mu(A) = \sup\{\ \mu(K) \ | \ K \subset A, \ K \in {\cal K}\ \},
\]
then $\mu$ is strongly regular.
\end{proposition}
%

In the rest of the paper, we assume that $(X, \ d)$ is complete
and separable metric space, and that $\mu$ is finite continuous fuzzy
measure.
In the following we show the main result in this paper.

%%%   Theorem 1   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}
  \label{Th01}
  If $\mu$ is null-additive, then $\mu$ is strongly regular.
\end{theorem}

To prove the theorem, we first prepare two lemmas.

%%%   Lemma 1   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}
  \label{lem-01}
Let $\mu$ be a finite continuous fuzzy measure. Then
for any $\epsilon > 0$ and any double sequence
$\{ A_{n}^{(k)} \mid n \geq 1, k \geq 1 \} \subset {\cal B}$
satisfying $A_{n}^{(k)} \searrow \emptyset \ (k \rightarrow \infty)$,
\ $n = 1,2,\ldots$,
there exists a subsequence $\{ A_{n}^{(k_{n})} \}$
of $\{ A_{n}^{(k)} \mid n \geq 1, k \geq 1 \}$
such that
\[
  \displaystyle{\mu \left(\bigcup_{n=1}^{\infty}A_{n}^{(k_{n})}
  \right)} < \epsilon \quad ( k_{1} < k_{2} < \ldots )
\]
\end{lemma}
{\bf Proof.} \
Since for any fixed $n =
1, 2, \ldots$, $A_{n}^{(k)} \searrow \emptyset$ as $k \rightarrow
\infty$,
for given $\epsilon > 0$, using
the continuity from above of fuzzy measures, we have $\lim_{k \to
+\infty} \mu(A_{1}^{(k)}) = 0$, therefore there
exists $k_1$ such that
$\mu(A_{1}^{(k_{1})}) < \frac{\epsilon}{2}$; For this $k_1$,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% the nextline changed
$
(A_{1}^{(k_{1})} \cup A_{2}^{(k)}) \searrow
A_{1}^{(k_{1})},
$
as $k \rightarrow \infty$. Therefore it follows, from the
continuity from above of $\mu$, that
$$
  \lim_{k \to +\infty} \mu(A_{1}^{(k_{1})} \cup A_{2}^{(k)}) = \mu(A_{1}^{(k_{1})}).
$$
Thus there exists $k_2$ ($> k_1$), such that
$$
  \mu(A_{1}^{(k_{1})} \cup A_{2}^{(k_{2})}) <
  \frac{\epsilon}{2}.
$$
%
Generally, there exist $k_1, k_2, \ldots, k_m$, such that
$$
  \mu(A_{1}^{(k_{1})} \cup A_{2}^{(k_{2})} \cup \ldots
  A_{m}^{(k_{m})}) < \frac{\epsilon}{2}.
$$
Hence we obtain a sequence $\{ k_n \}_{n = 1}^{\infty}$ of numbers
and a sequence $\{ A_{n}^{(k_{n})} \}_{n = 1}^{\infty}$ of  sets.
By using the monotonicity and the continuity from below of $\mu$,
we have
\[
  \mu\left(
     \bigcup_{n=1}^{+\infty} A_{n}^{(k_{n})}
     \right)
   \leq \frac{\epsilon}{2}
      < \epsilon.
\]
This gives the proof of the lemma.

%%%   Lemma 2   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}
  \label{lem-02}
If $\mu$ be continuous fuzzy measure, then for each $\epsilon > 0$,
 there exists a compact set $K_{\epsilon} \in {\cal K}$ such that
 $\mu(X - K_{\epsilon}) < \epsilon$.
\end{lemma}
{\bf Proof.} \ Since $(X, \ d)$ is separable, there exists a
countable dense subsets $\{x_i; i = 1,2,\ldots \}$. For
any for any $n, k \geq 1$, we put
\[
  \overline{S_k} (x_n) = \left\{ x \ : \ x \in X, \ d(x, x_n) \leq
  \frac{1}{k} \right\},
\]
then, for fixed $k=1, 2, \cdots$, as $m \to +\infty$
\[
  \bigcup_{n=1}^{m} \overline{S_k} (x_n) \nearrow
  \bigcup_{n=1}^{\infty} \overline{S_k} (x_n) = X.
\]
Thus, as $m \to +\infty$
\[
  X - \bigcup_{n=1}^{m} \overline{S_k}(x_n) \searrow \emptyset,
\]
for fixed $k=1, 2, \cdots$.
Applying Lemma 1 to the double sequence
 $\{X - \bigcup_{n=1}^{m} \overline{S_k}(x_n)\mid m \geq 1, k \geq 1 \}$,
then there exists a subsequence $\{m_k \}_k $
of the positive integers such that
\[
  \mu \left( \bigcup_{k=1}^{+\infty} \left( X - \bigcup_{n=1}^{m_k}
  \overline{S_k} (x_n) \right) \right) < \epsilon
\]
Put
\[
  K_{\epsilon} =
  \bigcap_{k=1}^{+\infty}  \bigcup_{n=1}^{m_k} \overline{S_k}(x_n).
\]
Thus, the closed set $K_{\epsilon}$ is totally bounded. From the completeness of $X$,
we know that $K_{\epsilon}$ is compact in $X$ and satisfies
\[
  \mu(X - K_{\epsilon})
  =
  \mu \left( \bigcup_{k=1}^{+\infty} \left( X - \bigcup_{n=1}^{m_k}
  \overline{S_k} (x_n) \right) \right) < \epsilon.
\]
Thus the lemma has proved. 

\bigskip

Now we will show the proof of Theorem~\ref{Th01} by using 
the previous lemmas.

\bigskip
\noindent 
{\bf Proof of Theorem~\ref{Th01}.} \
Let $A \in \cal{B}$ and given $\epsilon > 0$. From Proposition~\ref{Prop-a}
we know that $\mu$ is regular.
Therefore, there exist a sequence $\{ F^{(k)}
\}_{k=1}^{\infty}$ of closed sets and a sequence $\{ G^{(k)}
\}_{k=1}^{\infty}$ of open sets such that for every $k =
1,2,\ldots$, $F^{(k)} \subset A \subset G^{(k)}$,
\[
  \mu(G^{(k)} - F^{(k)}) < \frac{1}{k}.
\]
Without loss of generality, we can assume
that the
sequence $\{F^{(k)}\}_{k=1}^{\infty}$ is increasing in $k$ and the
sequence $\{G^{(k)}\}_{k=1}^{\infty}$ is decreasing in $k$.
Thus,
$\{G^{(k)} - F^{(k)}\}_{k=1}^{\infty}$ is a
decreasing sequence of sets with respect to $k$, and as $k
\rightarrow \infty$
\[
  G^{(k)} - F^{(k)} \searrow
  \displaystyle{\bigcap_{k=1}^{\infty}(G^{(k)} - F^{(k)})}.
\]
Denote $ D_{1} = \textstyle{\bigcap_{k=1}^{\infty}(G^{(k)} -
F^{(k)})}$, and noting that $\mu(D_{1}) \leq
\mu(G^{(k)} - F^{(k)}) < {1}/{k}, \ k = 1, 2,\ldots$, then $\mu(D_{1}) = 0$.

On the other hand, from Lemma~\ref{lem-02} there exists a sequence
$\{ K^{(k)} \}_{k=1}^{\infty}$ of compact subsets in $X$ such that
for every $k = 1,2,\ldots$
\[
  \mu(X - K^{(k)}) < \frac{1}{k},
\]
and we can assume that $\{K^{(k)}\}_{k=1}^{\infty}$ is decreasing in $k$.
Therefore, as $k \rightarrow \infty$
\[
  X - K^{(k)} \searrow
  \displaystyle{\bigcap_{k=1}^{\infty}(X - K^{(k)})}.
\]
Denote $ D_{1} = \textstyle{\bigcap_{k=1}^{\infty}(X - K^{(k)})}$,
then $\mu(D_{1}) = 0$. Thus, we have
\[
  (X - K^{(k)}) \cup (G^{(k)} - F^{(k)}) \searrow D_{1}\cup D_{2}
\]
as $k \rightarrow \infty$. Noting that $\mu(D_{1}\cup D_{2}) = 0$, by the continuity
of $\mu$, then
\[
  \lim_{k \to +\infty} \displaystyle{\mu
  \left((X - K^{(k)}) \cup (G^{(k)} - F^{(k)})\right)} = 0.
\]
Therefore there exists $k_{0}$ such that
\[
  \displaystyle{\mu
  \left((X - K^{(k_{0})}) \cup (G^{(k_{0})} - F^{(k_{0})})\right)} < \epsilon.
\]
Denoting $K_{\epsilon}=K^{(k_{0})} \cap F^{(k_{0})}$ and
$G_{\epsilon}=G^{(k_{0})}$, then $K_{\epsilon}$ is a compact set
and $G_{\epsilon}$ is an open set, and $K_{\epsilon} \subset A
\subset G_{\epsilon}$. Since $G_{\epsilon}-K_{\epsilon} \subset (X
- K^{(k_{0})}) \cup (G^{(k_{0})} - F^{(k_{0})})$, we obtain
\[
  \mu(G_{\epsilon}-K_{\epsilon})
  \leq \mu(X - K^{(k_{0})}) \cup (G^{(k_{0})} - F^{(k_{0})})
  < \epsilon.
\]
This shows that $\mu$ is strongly regular. {\it q.e.d.}

%%%   Corollary 1   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{corollary} \
 If $\mu$ is null-additive, then for any $A \in {\cal B}$
 the following statements hold:
 \begin{itemize}
    \item[{\rm (1)}] 
    \ For each $\epsilon > 0$,
  there exist a compact set $K_{\epsilon} \in {\cal K}$ such that
   $K_{\epsilon} \subset A$
  and $\mu(A - K_{\epsilon}) < \epsilon;$
    \item [{\rm (2)}]
 \
 $ \mu(A) = \sup\{\ \mu(K) \ | \ K \subset A, \ K \in {\cal K} \ \}$.
  \end{itemize}
\end{corollary}

By using the strongly regular of fuzzy measure, similar to 
the proof of Theorem 3 and 4 in \cite{Li04b}, we can prove 
the following theorems. 
They are a version of Egoroff's theorem and Lusin's
theorem on complete separable metric space, respectively.

%%%   Theorem 2   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem} \
 \label{Th02}
 {\rm (Egoroff's theorem)} \
Let $\mu$ be null-additive continuous fuzzy measure. If $\{f_n \}$
converges to $f$ almost everywhere on $X$, then for any $\epsilon
> 0$, there exists a compact subset $K_{\epsilon} \in \mathcal{K}$
 such that $\mu(X - K_{\epsilon}) < \epsilon$ and
$\{f_n \}_n$ converges to $f$ uniformly on $K_{\epsilon}$.
\end{theorem}

%%%   Theorem 3   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem} \
 \label{Th03}
{\rm (Lusin's theorem)} \
Let $\mu$ be null-additive continuous fuzzy measure.
 If $f$ is a real measurable function on $X$, then, for each $\epsilon > 0$,
there exists a compact subset $K_{\epsilon} \in \mathcal{K}$ such
that $f$ is continuous on $K_{\epsilon}$ and
 $\mu(X - K_{\epsilon}) \leq \epsilon.$
\end{theorem}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%       Section 3           
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Atoms of fuzzy measure}
\hspace{4mm}
In this section, as an application of strongly regularity, we shall
show a characterization of atom of null-additive fuzzy measure
on complete separable metric space.

%%%   Definition 4   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{definition}
  \label{df-atom}
  {\rm (\cite{Jiang96})
 \ A set $A \in {\mathcal{B}}$ with $\mu(A) > 0$ is call an atom
 if for any $B \subset A$ then
\begin{itemize}
  \item[(i)] \ $\mu(B) = 0$, or
  \item[(ii)] \ $\mu(A) = \mu(B)$ and $\mu(A - B) = 0$ holds.
\end{itemize}
}
\end{definition}
%

Consider a nonnegative real-valued measurable function $f$ on $A$.
The {\it fuzzy integral} of $f$ on $A$ with respect to $\mu$,
denoted by $(S) \int_{A} f d\mu$, is defined by
\[
(S) \int_{A} f d\mu = \displaystyle{\sup_{0 \leq \alpha <
+\infty}} \left[ \alpha \wedge \mu(\{ x : f(x) \geq \alpha \}\cap
A) \right]
\]

%%%   Theorem 4   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}
  \label{thm41} \
  Let $\mu$ be null-additive and continuous.
  If $A$ is an atom of $\mu$, then there exists a point
  $a \in A$ such that the fuzzy integral satisfies
$$
(S) \int_{A}fd\mu = f(a) \wedge \mu(\{ a \})
$$
for any non-negative measurable function $f$ on $A$.
\end{theorem}
{\bf Proof.} It is similar to the proof of Theorem 8 
in \cite{Jiang96}. {\it q.e.d.}

\bigskip
\noindent
{\bf Acknowledgement:}
The authors should express their thanks to referee who shows us
two references \cite{NaMu04} and \cite{Pap95b}. 

%**************************%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%
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%
\bibitem{Jiang96}
Q.~Jiang, H.~Suzuki, \ Fuzzy measures on metric spaces, {\it Fuzzy
Sets and Systems} {83}(1996) 99$-$106.
%
\bibitem{Li03b}
J.~Li, \ Order continuous of monotone set function and convergence
of measurable functions sequence, {\it Applied Mathematics and
Computation} \ {135}(2003) 211$-$218.
%
\bibitem{Li04b}
J.~Li, M.~Yasuda, \ Lusin's theorem on fuzzy measure spaces, {\it
Fuzzy Sets and Systems} 146(2004) 121$-$133.
%
\bibitem{Naru00}
Y.~Narukawa, T.~Murofushi, M.~Sugeno, \ Regular fuzzy measure and
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%
\bibitem{Naru03}
Y.~Narukawa, T.~Murofushi, \ Conditions for Choquet integral
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%
\bibitem{Naru04}
Y.~Narukawa, T.~Murofushi, \ Regular null-additive measure and
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%
\bibitem{NaMu04}
Y.~Narukawa, T.~Murofushi, 
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(2004), 517-521.
%
\bibitem{Pap95}
E.~Pap, {\it Null-additive Set Functions}, Kluwer, Dordrecht,
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%
\bibitem{Pap95b}
E.~Pap,  Regular null additive monotone set functions,
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%
\bibitem{Song03}
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%
\bibitem{Wu01}
J.~Wu, C.~Wu, \ Fuzzy regular measures on topological spaces, {\it
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%
\bibitem{Wang92}
Z.~Wang, G.~J.~Klir, {\it Fuzzy Measure Theory}, Plenum, New York,
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\end{thebibliography}

\end{document}

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\usepackage{latexsym,amssymb,theorem}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\newtheorem{prop}{Proposition}
\newtheorem{df}{Definition}
\newtheorem{thm}{Theorem}
\newtheorem{cor}{Corollary}
\newtheorem{lem}{Lemma}
\newtheorem{rem}{Remark}
\newtheorem{ex}{Example}
%
\begin{document}
%-------------------------the text-----------------------
