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\title{Further properties of null-additive fuzzy measure 
        on metric spaces}
\author{Jun Li${}^{(a)}$, Masami Yasuda${}^{(b)}$, Jinjie Song${}^{(c)}$}
\institution{${}^{(a)}$Communication University of China, Beijing, China\\
${}^{(b)}$Chiba University, Chiba, Japan\\
${}^{(c)}$Tianjin University of Technology, Tianjin, China
}
%\email{}
\slideCaption{MDAI2005}



\begin{document}

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%\bf\mathversion{bold}\maketitle


%% Text start %%

\maketitle

\begin{slide}{Objective:} %=================================
%{\bf Objective:}
Properties of
\textcolor{red}{ null-additive fuzzy measure}
on metric spaces

In this paper, under 
 \begin{itemize}
   \item the null-additivity, 
   \item weekly null-additivity and 
   \item converse null-additivity condition, 
 \end{itemize}
we shall discuss the relation among 
 \begin{itemize}
   \item the inner regularity, 
   \item the outer regularity and 
   \item the regularity of fuzzy measure. 
 \end{itemize}
\end{slide}



\begin{slide}{Regularity:} %=================================
{\bf Various regularities of set function:}

  %Pap95,
  %\bibitem{Pap95}
  $\bullet$ \textcolor{blue}{E.~Pap,} {\it Null-additive Set Functions}, 
  Kluwer, Dordrecht, 1995.

  %Jiang96,
  %\bibitem{Jiang96}
  $\bullet$ \textcolor{blue}{Q.~Jiang, H.~Suzuki,} 
  \ Fuzzy measures on metric spaces, 
  {\it Fuzzy Sets and Systems} {83}(1996) 99--106.

  %Wu01
  %\bibitem{Wu01}
  $\bullet$ \textcolor{blue}{J.~Wu, C.~Wu,} 
  \ Fuzzy regular measures on topological spaces, 
  {\it Fuzzy Sets and Systems} {119}(2001) 529--533.
\end{slide}


\begin{slide}{References(continued)} %=================================

  %Li04b, Song03,
  %\bibitem{Song03}
  $\bullet$ \textcolor{blue}{J.~Song, J.~Li,} 
  \ Regularity of null-additive fuzzy measure on
  metric spaces, {\it Int. J. General Systems} {32}(2003) 271--279.

  %\bibitem{Li03b}
  $\bullet$ \textcolor{blue}{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}
%$\bullet$ J.~Li, M.~Yasuda, \ Lusin's theorem on fuzzy measure spaces, {\it
%Fuzzy Sets and Systems} 146(2004) 121--133.


%\vspace{3ex}
%The regularity of null-additive fuzzy measure and proved Egoroff's theorem and
%Lusin's theorem for fuzzy measure on metric space.

\end{slide}

\begin{slide}{References(continued)} %=================================

  %Li04b, Song03,
  %\bibitem{Song03}
  $\bullet$ \textcolor{blue}{J.~Song, J.~Li,} 
  \ Regularity of null-additive fuzzy measure on
  metric spaces, {\it Int. J. General Systems} {32}(2003) 271--279.

  %\bibitem{Li03b}
  $\bullet$ \textcolor{blue}{J.~Li,} 
  \ Order continuous of monotone set function and convergence
  of measurable functions sequence, {\it Applied Mathematics and
  Computation} \ {135}(2003) 211--218.

\end{slide}

\begin{slide}{References(continued)} %=================================

  %\bibitem{Li04b}
  $\bullet$ \textcolor{blue}{J.~Li, M.~Yasuda,}
  \ Lusin's theorem on fuzzy measure spaces, {\it
  Fuzzy Sets and Systems} 146(2004) 121--133.

\vspace{3ex}\textcolor{yellow}{
The regularity of null-additive fuzzy measure and proved Egoroff's theorem and
Lusin's theorem for fuzzy measure on metric space.}

\end{slide}


\begin{slide}{References(continued)} %=================================

%Naru00,Naru03,Naru04,
%\bibitem{Naru00}
$\bullet$ \textcolor{blue}{Y.~Narukawa, T.~Murofushi, M.~Sugeno,}
 \ Regular fuzzy measure and
representation of comonotonically additive functional, {\it Fuzzy
Sets and Systems} 112(2000) 177--186.

%\bibitem{Naru03}
$\bullet$ \textcolor{blue}{Y.~Narukawa, T.~Murofushi,} 
\ Conditions for Choquet integral
representation of the comonotonically additive and monotone
functional, {\it J. Math. Anal. Appl.} 282(2003) 201--211.

%\bibitem{Naru04}
$\bullet$ \textcolor{blue}{Y.~Narukawa, T.~Murofushi,} 
\ Regular null-additive measure and
Choquet integral, {\it Fuzzy Sets and Systems} 143(2004)
487$-$492.
%
\end{slide}

\begin{slide}{General references:} %=================================

%\bibitem{Dobr74}
$\bullet$ \textcolor{blue}{I.~Dobrakov,} 
\ On submeasures I, {\it Dissertations Math.}
{112}(1974) 1-35.

%\bibitem{Wang92}
$\bullet$ \textcolor{blue}{Z.~Wang, G.~J.~Klir,} 
{\it Fuzzy Measure Theory}, Plenum, New York,
1992.

\end{slide}

\begin{slide}{Notation:} %=================================

$(X,  d)$ : a metric space, 

${\cal O}$ : the classes of all \textcolor{magenta}{open} sets in $(X, d)$ \\
${\cal C}$ : the classes of all \textcolor{magenta}{closed} sets in $(X, d)$ \\
${\cal K}$ : the classes of all \textcolor{magenta}{compact} sets in $(X, d)$ \\

${\cal B}$ denotes \textcolor{magenta}{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 ${\cal B}$

\end{slide}


\begin{slide}{Definition: continuity} %=================================
A set function $\mu:{\cal B} \to [0, +\infty]$ is said to be

 \textcolor{green}{{\it continuous from below}},
 if $\lim_{n \to \infty}\mu(A_n) = \mu(A)$
whenever $A_n \nearrow A$;

 \textcolor{yellow}{{\it continuous from above}}, if
 $\lim_{n \to \infty}\mu(A_n) = \mu(A)$
whenever $A_n \searrow A$;

 \textcolor{cyan}{{\it strongly order continuous}}, if
$\textstyle{\lim_{n \to \infty}} \mu(A_n) = 0$ whenever
$A_n \searrow \ B$ and $\mu(B) = 0$;

\end{slide}

\begin{slide}{Definition: additivities} %=================================
 \textcolor{green}{{\it null-additive}}, if
 $\mu(E \cup F) = \mu(E)$ for any $E$ whenever $\mu(F) = 0$;

 \textcolor{yellow}{{\it weakly null-additive}}, if
 $\mu(E \cup F) = 0$ whenever $ \mu(E) = \mu(F) = 0$;

 \textcolor{cyan}{{\it converse-null-additive}}, if
 $\mu(E - F) = 0$ whenever $F \subset E$ and $\mu(F) = \mu(E) <
 +\infty$;


\vspace{3ex}

Obviously, the null-additivity of $\mu$ implies weakly
null-additivity.
\end{slide}
%

\begin{slide}{Definition: a fuzzy measure} %=================================

{\rm
 \textcolor{magenta}{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{slide}

\begin{slide}{Definition: Wu/Wu {\it FSS}(2001)} %=========================

 A fuzzy measure $\mu$ is called
 \textcolor{blue}{{\it outer regular}} 
 if for each $A \in {\cal B}$ and each $\epsilon > 0$,
 there exists a set $G \in {\cal O}$ such that
 \textcolor{magenta}{$A \subset G, \ \mu(G - A) < \epsilon$}

 A fuzzy measure $\mu$ is called
 \textcolor{cyan}{{\it inner regular}},
 if for each $A \in {\cal B}$ and each $\epsilon > 0$,
 there exists a set $F \in {\cal C}$ such that
 \textcolor{green}{$F \subset A, \ \mu(A - F) < \epsilon$}


 A fuzzy measure $\mu$ is called 
 \colorbox{blue}{\textcolor{white}{\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 
 \textcolor{red}{$F \subset A \subset G$ 
 and $\mu(G - F) < \epsilon$}.

\end{slide}


\begin{slide}{Proposition: Li/Yasuda {\it FSS}(2004)} %=======================
If $\mu$ is weekly null-additive and continuous,
then it is regular. 

\vspace{3ex}
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{slide}


\begin{slide}{Proposition:} %=================================
If $\mu$ is weekly null-additive and strongly order continuous,
then both outer regularity and inner regularity imply regularity.
\end{slide}
%

\begin{slide}{Proposition:} %=================================
Let $\mu$ be a \textcolor{cyan}{null-additive} fuzzy measure.

 {\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 $A \in {\cal B};$

 {\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{slide}
%

\begin{slide}{Proposition:} %=================================
Let $\mu$ be a \textcolor{magenta}{converse-null-additive} 
fuzzy measure.

{\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.

{\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{slide}
%

\begin{slide}{Definition: strongly regular} %=================================

 A fuzzy measure $\mu$ is called 
 \textcolor{red}{{\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{slide}

%

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

\begin{slide}{Proposition:} %=================================
 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 \textcolor{red}{strongly regular}.
\end{slide}
%

\begin{slide}{} %=================================
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.
\end{slide}

\begin{slide}{Theorem:} %=================================
  If $\mu$ is null-additive, then $\mu$ is strongly regular.
\end{slide}


\begin{slide}{Lemma:} %=================================
 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{slide}



\begin{slide}{Lemma:} %=================================
 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{slide}


\begin{slide}{Corollary:} %=================================
 If $\mu$ is null-additive, then for any $A \in {\cal B}$
 the following statements hold:

 {\rm (1)} \ For each $\epsilon > 0$,
  there exist a compact set $K_{\epsilon} \in {\cal K}$ 
  such that
  $K_{\epsilon} \subset A$
  and $\mu(A - K_{\epsilon}) < \epsilon;$

 {\rm (2)} \
  $ \mu(A) = \sup\{\ \mu(K) \ | \ K \subset A, \ K \in {\cal K} \ \}$.
\end{slide}


\begin{slide}{Theorem: 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{slide}


%

\begin{slide}{Theorem: 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{slide}



\begin{slide}{Definition: Jiang/Suzuki {\it FSS}(1996)} %========
 A set $A \in {\mathcal{B}}$ with $\mu(A) > 0$ is call an 
 \textcolor{red}{atom}
 if $B \subset A$ then \par
 (i) \ $\mu(B) = 0$, or \par
 (ii) \ $\mu(A) = \mu(B)$ and $\mu(A - B) = 0$.
\end{slide}


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



\begin{slide}{Theorem:} %=================================
  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{slide}




\end{document}
