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%%         " Approximation of fuzzy neural networks        %%
%%                 by using Lusin's theorem"               %%
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%%                 J.Li, J.Li, M.Yasuda                    %%
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%%--in preparing for IPMU2006                              %%
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%%--revised version--2006/03/22---from Li Jun    
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%---------------------Text-------------------------------
\title{
Approximation of fuzzy neural networks\\  by \\
             using Lusin's theorem }
\author{
 Jun Li; 
  Commun.~Univ.~of China, China\\
 Jianzeng Li; 
  Commun.~Univ.~of China, China\\
 Masami Yasuda; 
  Chiba University, Japan\\
}
%
%
\begin{document}

\maketitle

\begin{slide}{Abstract}

  In this note, we study an approximation property of regular fuzzy
  neural network(RFNN). It is shown that any fuzzy-valued measurable
  function can be approximated by the four-layer RFNN in the sense
  of fuzzy integral norm for the finite sub-additive fuzzy measure
  on $\mathbb{R}$.

  \vspace{2mm}
  {\it Keywords}: 
  
  Fuzzy measure; Lusin's theorem; Approximation;
                 Regular fuzzy neural network
\end{slide}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%       Section 1           %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
\begin{slide}{Introduction 1/4}
  In neural network theory, the learning ability of a
  neural network is closely related to its approximating
  capabilities, so it is important and interesting to study the
  approximation properties of neural networks. 
\end{slide}

\begin{slide}{Introduction 2/4}
  The studies on this
  matter were undertaken by many authors and a great number of
  important results were obtained.
  
  $\bullet$ 
   J.G~Attali, G.~Pages, \ Approximation of functions by a multilayer
    perceptron: a new approach, {\it Neural Networks} 10(1997)
    1069-1081,
    
  $\bullet$ 
   R.M~Burton, H.G.~Dehling, \ Universal approximation in $p$-mean by
    neural networks, {\it Neural Networks} 11(1998) 661-667,
    
  $\bullet$ 
  F~Scarselli, A.G.~Tsoi, \ Universal approximation using
    feedforward neural networks: a survey of some existing methods,
    and some new results, {\it Neural Networks} 11(1998) 15-17.
\end{slide}

\begin{slide}{Introduction 3/4}
The similar approximation problems in fuzzy environment were
investigated by

  $\bullet$ 
  J.J~Buckley, Y.~Hayashi, \ Fuzzy input-output controllers are
   universal approximators, {\it Fuzzy Sets and Systems} 58(1993)
   273-278,

  $\bullet$ 
  J.J~Buckley, Y.~Hayashi, \ Can fuzzy neural nets approximate
   continuous fuzzy function, {\it Fuzzy Sets and Systems} 61(1994)
   43-51,

  $\bullet$ 
  P.~Liu, \ Analyses of regular fuzzy neural networks for
   approximation capabilities, {\it Fuzzy Sets and Systems} \
   114(2000) 329-338,

  $\bullet$ 
  P.~Liu, \ Universal approximations of continuous fuzzy-valued
   functions by multi-layer regular fuzzy neural networks, {\it Fuzzy
   Sets and Systems} \ 119(2001) 313-320. 
\end{slide}

\begin{slide}{Introduction 4/4}
In P.~Liu(2001)
proved that continuous fuzzy-valued function can be closely
approximated by a class of regular fuzzy neural networks (RFNNs)
with real input and fuzzy-valued output. 

In this note, by using
Lusin's theorem on fuzzy measure space, we show that such RFNNs is
pan-approximator for fuzzy-valued measurable function. 

That is,
any fuzzy-valued measurable function can be approximated by the
four-layer RFNNs in the sense of fuzzy integral norm for the
finite sub-additive measure on $\mathbb{R}$.
%
\end{slide}


%********************************
\section{Preliminaries}
% section 2

\begin{slide}{Preliminaries 1/3}
We suppose that $(X, \rho)$ is a metric space, and
that ${\cal O}$ and ${\cal C}$ are the classes of all open and
closed sets in $(X, \rho)$, respectively, and ${\cal B}$ is Borel
$\sigma$-algebra on $X$, i.e., it is the smallest $\sigma$-algebra
containing ${\cal O}$.

A set function $\mu:{\cal B} \to [0, +\infty)$ is
called
 {\it a fuzzy measure}(Narukawa/Murofushi(2004)),
 if it satisfies the following properties:

\begin{description}
 \item[(FM1)]\ $\mu(\emptyset)=0$; 
 \item[(FM2)]\ $A \subset B$ \ implies \
  $\mu(A) \leq \mu(B)$. 
\end{description}

A fuzzy measure $\mu$ is called
 {\it null-additive} (Wang/Klir(1992)), if for any
$E, F \in {\cal B}$ and $\mu(F) = 0$ imply
 $\mu(E \cup F) = \mu(E)$;
 {\it sub-additive} (Pap(1995)), if
for any $E, F \in {\cal B}$ we have
 $\mu (E \cup F) \leq \mu(E)+\mu(F)$.
%

\end{slide}

%\begin{emptyslide}{Insert Graphics}
%\centering
%\vspace{\stretch{1}}
%\rotatebox{270}{\includegraphics[height=0.8\slideheight]{4layer.eps}}
%\vspace{\stretch{1}}
%\end{emptyslide}

%\includegraphics[width=100mm]{4layer.eps}


\begin{slide}{Preliminaries 2/3}
In this paper, we always assume that $\mu$ is a
finite, sub-additive and continuous fuzzy measure on ${\cal B}$.

Consider a nonnegative real-valued measurable
function $f$ on $A$ and the {\it fuzzy integral} of $f$ on $A$
with respect to $\mu$, which is denoted by
% by $\displaystyle (S)\int f \, d\mu$, is defined by
\[
  \begin{array}{ll}
    & (S)\int_{A} f \, d\mu \\[2ex]
   \triangleq  & \displaystyle{\sup_{0 \leq \alpha <
   + \infty}}
   \left[ \alpha \wedge \mu(\{ x : f(x) \geq \alpha \}
     \cap A) \right]
  \end{array}
\]
%
\end{slide}


\begin{slide}{Preliminaries 3/3}
\begin{thm}{\rm(Lusin's theorem (Li/Yasuda(2004), Song/Li(2003))}
 \label{Lusin's theorem}

 %theorem-01} \
 Let $(X, \rho)$ be metric space and $\mu$ be null additive fuzzy
 measure on ${\cal B}$.
 If $f$ is a real-valued measurable function on $E \in {\cal B}$,
 then, for every $\epsilon > 0$,
 there exists a closed subset $F_{\epsilon} \in {\cal B}$
 such that $f$ is continuous on $F_{\epsilon}$
 and $\mu(E - F_{\epsilon}) < \epsilon.$
\end{thm}
%
\end{slide}


%*****************************************
\section{Approximation in fuzzy mean by RFNN}
%Approximation in fuzzy mean by regular fuzzy neural networks
% section 3

\begin{slide}{Approximation 1/10}
In this section, we study an approximation property of
the four-layer RFNNs to fuzzy-valued measurable function 
in the sense of fuzzy integral norm for fuzzy measure on 
$\mathbb{R}$.
\end{slide}


\begin{slide}{Approximation 2/10}
Let ${\mathcal{F}}_{0}(\mathbb{R})$ be the set of all
bounded fuzzy numbers, i.e., for $\tilde{A} \in
{\mathcal{F}}_{0}(\mathbb{R})$, the following conditions hold:
\begin{itemize}
 \item[(i)]  $\forall \ \alpha \in (0, 1],
   \tilde{A}_{\alpha}
      \triangleq \{x \in \mathbb{R} \mid \tilde{A}(x) \geq \alpha\}$
   is the closed interval of $\mathbb{R}$;
 \item[(ii)]  The support Supp$(\tilde{A})
   \triangleq \mbox{cl}\{x \in \mathbb{R} \mid \tilde{A}(x) > 0\}
   \subset  $ is a bounded set;
 \item [(iii)]  $\{x \in \mathbb{R} \mid \tilde{A}(x) = 1\}\neq \emptyset$.
\end{itemize}

For simplicity, supp($\tilde{A}$) is also written as
$\tilde{A_{0}}$. Obviously, $\tilde{A_{0}}$ is a bounded and closed
interval of $\mathbb{R}$. For $\tilde{A}\in
{\mathcal{F}}_{0}(\mathbb{R})$, let $\tilde{A}_{\alpha} =
[a_{\alpha}^{-}, a_{\alpha}^{+}]$ for each $\alpha\in [0,1]$ and we
denote
$$
  |\tilde{A}|
      \triangleq
      \bigvee_{\alpha\in[0, 1]}(|a_{\alpha}^{-}|\vee|a_{\alpha}^{+}|).
$$
\end{slide}


\begin{slide}{Approximation 3/10}
\begin{prop}\label{Prop-00} {Liu(2001)}
%Prop3.1
\
  Assume $\tilde{A}, \tilde{A_{1}}, \tilde{A_{2}} \in
  {\mathcal{F}}_{0}(\mathbb{R})$, and $\tilde{W_{i}}, \tilde{V_{i}}
  \in {\mathcal{F}}_{0}(\mathbb{R})( i=1,2,\cdots,n)$. 
  
  Then
\begin{itemize}
  \item[{\rm (1)}] \ $d(\tilde{A}\cdot\tilde{A_{1}},
   \tilde{A}\cdot\tilde{A_{2}})
   \leq |\tilde{A}|\cdot d(\tilde{A_{1}}, \tilde{A_{2}})$, \\[1.5ex]
  \item[{\rm (2)}] \ $d(\displaystyle\sum_{i=1}^{n}\tilde{W}_{i},
    \sum_{i=1}^{n}\tilde{V}_{i})
  \leq \sum_{i=1}^{n}d(\tilde{W}_{i}, \tilde{V}_{i})$.
 \end{itemize}
\end{prop}

\end{slide}


\begin{slide}{Approximation 4/10}
For $\tilde{A}, \tilde{B} \in {\mathcal{F}}_{0}(\mathbb{R})$,
define metric $d(\tilde{A}, \tilde{B})$ between $\tilde{A}$ and
$\tilde{B}$ by
\[
  d(\tilde{A}, \tilde{B}) \triangleq \displaystyle{\sup_{\alpha\in
  [0,1]}d_{H}(\tilde{A}_{\alpha}, \tilde{B}_{\alpha})}
\]
where $d_{H}$ means Hausdorff metric: for $A, B \subset
\mathbb{R}$,
\[
  \begin{array}{ll}
             &  d_{H}(A, B) \\[2ex]
   \triangleq \! \!  &  \! \! \max \left\{
            \displaystyle\sup_{x\in A}\inf_{y\in B}(|x-y|),%\right. \\[2ex]
             %&  \qquad \, \, \left.
             \displaystyle\sup_{y\in B}\inf_{x\in A}(|x-y|) \right\} .
 \end{array}
\]

\end{slide}


\begin{slide}{Approximation 5/10}
It is known that $({\mathcal{F}}_{0}(\mathbb{R}), d)$
is a completely separable metric space (Diamond/Kloeden(1994)).

Let $T$ be a measurable set in $\mathbb{R}^{n}$, $(T,
\mathcal{B}\cap T, \mu)$ finite fuzzy measure space. Let
$\mathcal{L}(T)$ denote the set of all fuzzy-valued measurable
function
 \[
 \tilde{F}: T \rightarrow
{\mathcal{F}}_{0}(\mathbb{R}).
\]

For any $\tilde{F_{1}}, \tilde{F_{2}} \in
\mathcal{L}(T)$, $d(\tilde{F_{1}}, \tilde{F_{2}})$ is measurable
function on $(T, \mathcal{B}\cap T)$, we will write a fuzzy
integral norm as
\[
{\bigtriangleup}_{S}(\tilde{F_{1}}, \tilde{F_{2}})
 \triangleq (S) \int_{T} d(\tilde{F_{1}}, \tilde{F_{2}}) d\mu.
\]
%

\end{slide}


\begin{slide}{Approximation 6/10}
\begin{prop} \label{Prop-01}
%Prop 3.2
\
Let $\tilde{F_{1}}, \tilde{F_{2}}, \tilde{F_{3}} \in
\mathcal{L}(T)$, then
\[
{\bigtriangleup}_{S}(\tilde{F_{1}}, \tilde{F_{3}}) \leq 2
({\bigtriangleup}_{S}(\tilde{F_{1}}, \tilde{F_{2}}) +
{\bigtriangleup}_{S}(\tilde{F_{2}}, \tilde{F_{3}}) ).
\]
\end{prop}
\end{slide}


\begin{slide}{Approximation 7/10}
{\bf Proof.} From subadditivity of $\mu$, we have
\begin{eqnarray*}
 & &{\bigtriangleup}_{S}(\tilde{F_{1}}, \tilde{F_{3}})
       = (S) \int_{T} d(\tilde{F_{1}}, \tilde{F_{3}}) d\mu
                                                    \\[2mm]
   & = &  \bigvee_{\alpha\in[0, \infty)}
         \{ \alpha \wedge\mu(T\cap(d(\tilde{F_{1}},
         \tilde{F_{3}}))_{\alpha} \}\\[2mm]
   & \leq & \bigvee_{\alpha\in[0, \infty)}
            \{ \alpha \wedge\mu(T \cap  \\[2mm]
   & &  (d(\tilde{F_{1}},
         \tilde{F_{2}})_{\frac{\alpha}{2}}\cup d(\tilde{F_{2}},
         \tilde{F_{3}})_{\frac{\alpha}{2}}) ) \} \\[2mm]
   &\leq& \bigvee_{\alpha\in[0, \infty)}
            \{ \alpha \wedge[ \mu(T\cap d(\tilde{F_{1}},
         \tilde{F_{2}})_{\frac{\alpha}{2}}) \\[2mm]
     & &  + \mu(T\cap d(\tilde{F_{2}},
         \tilde{F_{3}})_{\frac{\alpha}{2}}) ] \}.
\end{eqnarray*}

\end{slide}


%\begin{slide}{Approximation 3}
Because of the elementary inequality:
$
  a \wedge(b+c) \leq (a\wedge b) + (a \wedge c)
$
where $a,b,c \geq 0$, 
%\end{slide}


\begin{slide}{Approximation 8a/10}
we have
\begin{eqnarray*}
  {\bigtriangleup}_{S}(\tilde{F_{1}}, \tilde{F_{3}})
  &\leq & \bigvee_{\alpha\in[0, \infty)}
            \{ \alpha \wedge \mu(T\cap d(\tilde{F_{1}},
         \tilde{F_{2}})_{\frac{\alpha}{2}}) \\[2mm]
  & &  + \alpha \wedge \mu(T\cap d(\tilde{F_{2}},
         \tilde{F_{3}})_{\frac{\alpha}{2}})  \} \\[2mm]
  &\leq & \bigvee_{\alpha\in[0, \infty)}
            [ \alpha \wedge \mu(T\cap d(\tilde{F_{1}},
         \tilde{F_{2}})_{\frac{\alpha}{2}})] \\[2mm]
  & &  + \bigvee_{\alpha\in[0, \infty)}
            [ \alpha \wedge \mu(T\cap d(\tilde{F_{2}},
         \tilde{F_{3}})_{\frac{\alpha}{2}})  ] \\[2mm]
\end{eqnarray*}
\end{slide}

\begin{slide}{Approximation 8b/10}
\begin{eqnarray*}         
  &\leq & \bigvee_{\alpha\in[0, \infty)}
            \left[ \frac{\alpha}{2} \wedge \mu(T\cap d(\tilde{F_{1}},
         \tilde{F_{2}})_{\frac{\alpha}{2}})\right. \\[2mm]
  & &  + \left. \frac{\alpha}{2} \wedge \mu(T\cap d(\tilde{F_{1}},
         \tilde{F_{2}})_{\frac{\alpha}{2}})\right] \\[2mm]
  & &  + \bigvee_{\alpha\in[0, \infty)}
            \left[ \frac{\alpha}{2} \wedge \mu(T\cap d(\tilde{F_{2}},
         \tilde{F_{3}})_{\frac{\alpha}{2}})\right. \\[2mm]
  & &  +  \left. \frac{\alpha}{2} \wedge \mu(T\cap d(\tilde{F_{2}},
         \tilde{F_{3}})_{\frac{\alpha}{2}})\right] \\[2mm]
\end{eqnarray*}
\end{slide}

\begin{slide}{Approximation 8c/10}
\begin{eqnarray*}         
  &\leq & \bigvee_{\frac{\alpha}{2}\in[0, \infty)}
          \left[ \frac{\alpha}{2} \wedge \mu(T\cap d(\tilde{F_{1}},
         \tilde{F_{2}})_{\frac{\alpha}{2}})\right] \\[2mm]
  & &  +  \bigvee_{\frac{\alpha}{2}\in[0, \infty)}
          \left[ \frac{\alpha}{2} \wedge \mu(T\cap d(\tilde{F_{1}},
         \tilde{F_{2}})_{\frac{\alpha}{2}})\right] \\[2mm]
  & &  + \bigvee_{\frac{\alpha}{2}\in[0, \infty)}
          \left[ \frac{\alpha}{2} \wedge \mu(T\cap d(\tilde{F_{2}},
         \tilde{F_{3}})_{\frac{\alpha}{2}})\right] \\[2mm]
  & &  +  \bigvee_{\frac{\alpha}{2}\in[0, \infty)}
         \left[ \frac{\alpha}{2} \wedge \mu(T\cap d(\tilde{F_{2}},
         \tilde{F_{3}})_{\frac{\alpha}{2}})\right] \\[2mm]
  &=&  2\left({\bigtriangleup}_{S}(\tilde{F_{1}}, \tilde{F_{2}}) +
{\bigtriangleup}_{S}(\tilde{F_{2}}, \tilde{F_{3}}) \right).
\end{eqnarray*}
%
\end{slide}


\begin{slide}{Approximation 9/10}
\begin{df}(Liu(2001)) \
A fuzzy-valued function 
$\tilde{\Phi}: T \rightarrow {\mathcal{F}}_{0}(\mathbb{R})$ 
is called a fuzzy-valued simple function, 
if there exist $\tilde{A}_{1}, \tilde{A}_{2}, \ldots, 
\tilde{A}_{m} \ \in {\mathcal{F}}_{0}(\mathbb{R})$, 
such that $\forall \ x \in T$,
 \[
 \tilde{\Phi}(x) =
 \sum_{k=1}^{m}\tilde{A}_{k}\cdot\chi_{T_{k}}(x)
 \]
 where
  $T_{k} \in \mathcal{B}\cap T \ (k=1,2,\ldots,m), \
  T_{i}\cap T_{j} = \emptyset \ (i \neq j)$ and $T
  =\bigcup_{k=1}^{m}T_{k}$.
\end{df}

\end{slide}


\begin{slide}{Approximation 10/10}
Immediately, if $\mathcal{S}(T)$ denotes the set of
all fuzzy-valued simple functions, then $\mathcal{S}(T) \subset
\mathcal{L}(T)$.

Similar to the proof of Proposition \ref{Prop-01} and
by using subadditivity of $\mu$, we can obtain the following
proposition.

\begin{prop} \label{Prop-02} \
Let $\mu$ be a finite, sub-additive and continuous fuzzy measure
on $\mathbb{R}$. If $ \tilde{F} \in \mathcal{L}(T)$, then for
every $\epsilon > 0$, there exists $\tilde{\Phi}_{\epsilon} \in
\mathcal{S}(T)$ such that
 \[{\bigtriangleup}_{S}(\tilde{F}, \tilde{\Phi}_{\epsilon})<
\epsilon.
 \]
\end{prop}
\end{slide}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% new section 4
\section{Main Theorem}

\begin{slide}{Main Theorem 1/6}
Define
\[
   \mathcal{H}[\sigma] \triangleq
   \left\{ \tilde{H} \, \Biggm| \, \tilde{H}(x) =
   \sum_{i=1}^{n}\tilde{W}_{i} \,
      V_i[\sigma]
      %    \left(\sum_{j=1}^{m}\tilde{V}_{ij} \cdot
      %           \sigma(x\cdot \tilde{U}_{j} + \tilde{\Theta}_{j})
      %    \right)
   \right\}
\]
where
\[
     V_i[\sigma]  \triangleq  \sum_{j=1}^{m}\tilde{V}_{ij} \cdot
                 \sigma(x\cdot \tilde{U}_{j} + \tilde{\Theta}_{j})
 \]
and
$\sigma$ is a given extended function of
$\sigma : \mathbb{R} \rightarrow \mathbb{R}$
(bounded, continuous and nonconstant), and
$x \in \mathbb{R}, \tilde{W}_{i},
\tilde{V}_{ij}, \tilde{U}_{j}, \tilde{\Theta}_{j}
\in {\mathcal{F}}_{0}(\mathbb{R})$.

\end{slide}


\begin{slide}{Main Theorem 2/6}
For any $\tilde{H} \in \mathcal{H}[\sigma]$,
$\tilde{H}$ is a four-layer feedforward RFNN with activation
function $\sigma$, threshold vector $(\tilde{\Theta}_{1}, \ldots,
\tilde{\Theta}_{m})$ in the first hidden layer(cf. Liu(2001).

\end{slide}


\begin{slide}{Main Theorem 3/6}
Restricting fuzzy numbers $\tilde{V}_{ij},
\tilde{U}_{j}, \tilde{\Theta}_{j} \in
{\mathcal{F}}_{0}(\mathbb{R})$, respectively, to be real numbers
$v_{ij}, u_{j}, \theta_{j} \in \mathbb{R}$, we obtain the subset
$\mathcal{H}_{0}[\sigma]$ of $\mathcal{H}[\sigma]$:
\[
\mathcal{H}_{0}[\sigma] \triangleq \left\{ \tilde{H} \, \Biggm| \, \tilde{H}(x) =
\sum_{i=1}^{n}\tilde{W}_{i} \,
          v_i[\sigma]
          %\left(\sum_{j=1}^{m}v_{ij} \cdot
          %       \sigma(x\cdot u_{j} + \theta_{j})
          %\right)
\right\}.
\]
where
\[
      v_i[\sigma] \triangleq
          \sum_{j=1}^{m}v_{ij} \cdot
                \sigma(x\cdot u_{j} + \theta_{j}).
\]
%
\end{slide}


\begin{slide}{Main Theorem 4/6}

Let define two classes of pan-approximation which is 
fundamental to our results. 
\begin{df}
\begin{itemize}
  \item[{\rm (1)}] \  $\mathcal{H}_{0}[\sigma]$ is call the
   pan-approximator of $\mathcal{S}(T)$ in the sense of
   ${\triangle}_{S}$, if for $\forall \ \tilde{\Phi}\in
   \mathcal{S}(T)$, $\forall \ \epsilon > 0$, there exists
   $\tilde{H}_{\epsilon}\in \mathcal{H}_{0}[\sigma]$ such that
   $
     {\triangle}_{S}(\tilde{\Phi}, \tilde{H}_{\epsilon}) < \epsilon.
   $
  \item[{\rm (2)}] \  For $\tilde{F}\in \mathcal{L}(T)$,
   $\mathcal{H}[\sigma]$ is call the pan-approximator for
   $\tilde{F}$ in the sense of ${\triangle}_{S}$, if $\forall \
   \epsilon > 0$, there exists $\tilde{H}_{\epsilon} \in
   \mathcal{H}[\sigma]$ such that
   $
    {\triangle}_{S}(\tilde{F}, \tilde{H}_{\epsilon}) < \epsilon.
   $
\end{itemize}
\end{df}
%
\end{slide}


\begin{slide}{Main Theorem 5/6}
By using Lusin's theorem (Theorem \ref{Lusin's
theorem}), Proposition \ref{Prop-01} and \ref{Prop-02} we can
obtain the main result in this paper, which is stated in the
following.
%
\end{slide}


\begin{slide}{Main Theorem 6/6}
\begin{thm}% Theorem 3.1
Let $(T, \mathcal{B}\cap T, \mu)$ be fuzzy measure space and $\mu$
be finite, sub-additive and continuous. Then,

\begin{itemize}
  \item[{\rm (1)}]  $\mathcal{H}_{0}[\sigma]$ is the pan-approximator of
    $\mathcal{S}(T)$
    in the sense of ${\triangle}_{S}$. \\[1.5ex]
  \item[{\rm (2)}]
    $\mathcal{H}[\sigma]$ is the
    pan-approximator for $\tilde{F}$ in the sense of
    ${\triangle}_{S}$.
\end{itemize}
\end{thm}
\end{slide}


\section{Pf of Main Theorem}

\begin{slide}{Pf of Main Thm 1/13}
{\bf Proof.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%---------------the proof of Theorem 3.1-------------------------
%{\bf Proof.} 
By using the conclusion of (1) and Proposition
\ref{Prop-02} we can obtain (2). Now we only prove (1).
Suppose that $\tilde{\Phi}(x)$ is a fuzzy-valued simple function,
i.e.,
$$
 \tilde{\Phi}(x) =
 \sum_{k=1}^{m}\chi_{T_{k}}(x)\cdot\tilde{A}_{k} \ \ (x \in T).
$$
For arbitrarily given $\epsilon > 0$,
applying Theorem \ref{Lusin's theorem} (Lusin's theorem)
to each real measurable function $\chi_{T_{k}}(x)$, for every 
fixed $k\ (1\leq k\leq m)$, there exists closed set $F_{k} \in 
\mathcal{B}\cap T$ such that
$$
 F_{k} \subset L_{k}
     \quad {\rm and} \quad
 \mu(L_{k} - F_{k}) < \frac{\epsilon}{2m}
$$
and $\chi_{T_{k}}(x)$ is continuous on $F_{k}$.
\end{slide}


\begin{slide}{Pf of Main Thm 2/13}
Therefore, for every $k$ there exist a Tauber-Wiener function 
$\sigma$ and $p_{k}\in N, v^{'}_{k1}, v^{'}_{k2}, \cdots, 
v^{'}_{kp_{k}}$, $\theta^{'}_{k1},\theta^{'}_{k2}, \cdots, 
\theta^{'}_{kp_{k}} \in \mathbb{R}$,
and $\mathbf{w}^{'}_{k1}, \mathbf{w}^{'}_{k2}, \cdots,
\mathbf{w}^{'}_{kp_{k}} \in \mathbb{R}^{n}$ such that
\[
 \biggm|\chi_{T_{k}}(x) - \sum_{j=1}^{p_{k}}v^{'}_{kj} \cdot
         \sigma(\langle \mathbf{w}^{'}_{kj}, x\rangle +
                                         \theta^{'}_{kj})\biggm|
   < \frac{\epsilon}{2\displaystyle\sum_{k=1}^{m}|\tilde{A}_{k}|} 
\]
for $ x \in L_{k}$.
Note that we can assume $\sum_{k=1}^{m}|\tilde{A}_{k}|\neq 0$, 
without any loss of generality. 
\end{slide}


\begin{slide}{Pf of Main Thm 3/13}
Denote $L = \bigcap_{k=1}^{m}L_{k}$, then $T =
L\cup(T-L)$. By the subadditivity of $\mu$, we have
\[
 \begin{array}{ll}
    \mu(T-L)   
   & = \mu(\bigcup_{k=1}^{m}(T-L_{k}))  \\[1.5ex] 
   & \leq
        \sum_{k=1}^{m}\mu(T-L_{k}) 
     <  \displaystyle\frac{\epsilon}{2}.
 \end{array}
\]
We take $\beta_{1} = 0, \ \beta_{k}=\sum_{i=1}^{k-1}p_{i}, k=2,\cdots, m$,
and $p=\sum_{k=1}^{m}p_{k}$. For $k=1,2,\cdots,m, j=1,2,\cdots,p$, we denote

\[
v_{kj}= \left\{
\begin{array}{ll}
   v^{'}_{k(j-\beta_{k})}, & \ \ \mbox{if} \ \ \beta_{k} < j \leq \beta_{k+1}, \\[2mm]
    0                      & \ \ \mbox{otherwise},  \\
\end{array} \right.
\]
\end{slide}


\begin{slide}{Pf of Main Thm 4/13}
%
\[
\theta_{kj}= \left\{
\begin{array}{ll}
   \theta^{'}_{k(j-\beta_{k})}, & \ \
                  \mbox{if} \ \ \beta_{k} < j \leq \beta_{k+1}, \\[2mm]
    0                           & \ \ \mbox{otherwise},  \\
\end{array} \right.
\]
%
\[
  \mathbf{w}_{kj}= \left\{
  \begin{array}{ll}
   \mathbf{w}^{'}_{k(j-\beta_{k})}, & \ \
                  \mbox{if} \ \ \beta_{k} < j \leq \beta_{k+1}, \\[2mm]
    0                           & \ \ \mbox{otherwise},  \\
  \end{array} \right.
\]
then, for any $k \in \{1,2,\cdots,m\}$, we have
\[
  \begin{array}{l}
  \sum_{j=1}^{p}v_{ij} \cdot
                 \sigma(\langle \mathbf{w}_{kj}, x\rangle +
                                         \theta_{kj}) \\[1.5ex]
  = \sum_{j=1}^{p_{k}}v^{'}_{ij} \cdot
                 \sigma(\langle \mathbf{w}^{'}_{kj}, x\rangle +
                                         \theta^{'}_{kj}).
  \end{array}
\]

\end{slide}


\begin{slide}{Pf of Main Thm 5/13}
Now denote that 
\[
  \tilde{H}(x) =
  \sum_{k=1}^{m}\tilde{A}_{k}\cdot
          \left(\sum_{j=1}^{p}v_{kj} \cdot
                 \sigma(\langle \mathbf{w}_{kj}, x\rangle +
                                         \theta_{kj})
          \right),
\]
then $\tilde{H}\in \mathcal{H}_{0}[\sigma]$. 

In the reminder part of this section we will prove ${\bigtriangleup}_{S}(\tilde{H},
\tilde{\Phi}) < \epsilon$.
%
\end{slide}


\begin{slide}{Pf of Main Thm 6/13}
Denote 
$$
   B_{kj} = v_{kj} \cdot
           \sigma(\langle \mathbf{w}_{kj}, 
           x\rangle + \theta_{kj})
$$
and
$$
  B_{kj}^{'} = v^{'}_{ij} \cdot
               \sigma(\langle \mathbf{w}^{'}_{kj}, 
               x\rangle + \theta^{'}_{kj}).
$$
By using Proposition \ref{Prop-00} and noting
 $\mu(T-L) < {\epsilon}/{2}$, we have
\begin{eqnarray*}
   & & {\bigtriangleup}_{S}(\tilde{H}, \tilde{\Phi})\\[2mm]
   &=& (S) \int_{T} d(\tilde{H}, \tilde{\Phi}) d\mu \\[2mm]
   &=& \bigvee_{0 \leq \alpha < + \infty}
          \left[
            \alpha \wedge \mu(T \cap (d(\tilde{H}, \tilde{\Phi}))_{\alpha})
          \right].
\end{eqnarray*}
\end{slide}


\begin{slide}{Pf of Main Thm 7/13}
Since 
\begin{eqnarray*}
 & & \mu
     ( T \cap 
        d(\tilde{H}, \tilde{\Phi})_{\alpha}
     ) \\[2mm]
 &=& \mu
     \left( T \cap 
          d\left(
            \sum_{k=1}^{m}\tilde{A}_{k}\cdot
            \sum_{j=1}^{p}B_{kj}, \right.\right. \\[2mm]
 & & %\hspace{7em} 
     \left.\left. 
            \sum_{k=1}^{m}\chi_{T_{k}}(x)\cdot
            \tilde{A}_{k}
          \right)_{\alpha}  
     \right) \\[2mm]
&\leq & 
         \mu\left(
          (L\cup(T-L)) \cap
          \left(
           C_{mp}
          \right)_{\alpha}
          \right) \\[2mm]
&\leq & 
         \mu\left(L \cap
          (
          C_{mp}
          )_{\alpha}
          \right)
  +   
         \mu\left((T-L) \cap
          (C_{mp})_{\alpha}
         \right)
\end{eqnarray*}

\end{slide}


\begin{slide}{Pf of Main Thm 8/13}

where the notation of set $C_{mp}$ 
is assigned as 
$$
  C_{mp} = 
          \sum_{k=1}^{m}|\tilde{A}_{k}|\cdot
          d\left(\sum_{k=1}^{p}B_{kj}, 
          \chi_{T_{k}}(x)\right).
$$
\end{slide}


\begin{slide}{Pf of Main Thm 9/13}
Hence
${\bigtriangleup}_{S}(\tilde{H}, \tilde{\Phi})$ is
dominated by
\begin{eqnarray*}
 & & \bigvee_{0 \leq \alpha < + \infty}
     \left[\alpha \wedge
         \mu\left(L \cap
          \left(
           D_{mp}
          \right)_{\alpha}
          \right)
     \right] \\
 & & + \bigvee_{0 \leq \alpha < + \infty}
     \left[\alpha \wedge
         \mu\left((T-L)
          \right)
     \right] \\[2mm]
 & \leq & \bigvee_{0 \leq \alpha < + \infty}
     \left[\alpha \wedge
         \mu\left(L \cap
          \left(
          D'_{mp}
          \right)_{\alpha}
          \right)
     \right] + \frac{\epsilon}{2}
\end{eqnarray*}

\end{slide}


\begin{slide}{Pf of Main Thm 10/13}

Here, for simplicity, we use the notations of
$$
   D_{mp} = \sum_{k=1}^{m}|\tilde{A}_{k}|\cdot
           \left|\sum_{j=1}^{p}B_{kj} -\chi_{T_{k}}(x)
           \right|
$$
and
$$
   D'_{mp} = \sum_{k=1}^{m}|\tilde{A}_{k}|\cdot
             d\left(
             \sum_{j=1}^{p_{k}}B_{kj}^{'} -     
             \chi_{T_{k}}(x)\right).
$$ 

\end{slide}


\begin{slide}{Pf of Main Thm 11/13}
Now we estimate the first part in the above formula. If $x\in L $,
then for every $k=1,2,\cdots,m$, we have $x\in L_{k}$, hence
\[
\biggm|\chi_{T_{k}}(x) - \sum_{j=1}^{p_{k}}v^{'}_{kj} \cdot
         \sigma(\langle \mathbf{w}^{'}_{kj}, x\rangle +
                                         \theta^{'}_{kj})\biggm|
   < \frac{\epsilon}{2\displaystyle\sum_{k=1}^{m}|\tilde{A}_{k}|},
\]
for every $k=1,2,\cdots,m$. That is, for $x\in L$,
\[
  D'_{mp} = \sum_{k=1}^{m}|\tilde{A}_{k}|\cdot
          d\left(\sum_{j=1}^{p_{k}}B_{kj}^{'} 
   - \chi_{T_{k}}(x)\right)
   < \frac{\epsilon}{2}.
\]

\end{slide}


\begin{slide}{Pf of Main Thm 12/13}

Therefore,
\begin{eqnarray*}
 & & \bigvee_{0 \leq \alpha < + \infty}
     \left[\alpha \wedge
         \mu\left(L \cap
          \left(
           D'_{mp}
          \right)_{\alpha}
          \right)
     \right] \\[2mm]
 &=&  \bigvee_{\alpha \in [0, \frac{\epsilon}{2}]}
     \left[\alpha \wedge
         \mu\left(L \cap
          \left(
           D'_{mp}
          \right)_{\alpha}
          \right)
     \right] \\[2mm]
  & &  +  \bigvee_{\alpha \in [\frac{\epsilon}{2}, \infty)}
     \left[\alpha \wedge
         \mu\left(L \cap
          \left(
           D'_{mp}
          \right)_{\alpha}
          \right)
     \right] \\[2mm]
  &=&  \bigvee_{\alpha \in [0, \frac{\epsilon}{2}]}
     \left[\alpha \wedge
         \mu\left(L \cap
          \left(
           D'_{mp}
          \right)_{\alpha}
          \right)
     \right] \\[2mm]
   &\leq& \frac{\epsilon}{2}.
\end{eqnarray*}
\end{slide}


\begin{slide}{Pf of Main Thm 13/13}
Thus, combining with the previous evaluation, we obtain
$$
\begin{array}{ll}
     & {\bigtriangleup}_{S}(\tilde{H}, \tilde{\Phi}) \\
  \leq & \bigvee_{0 \leq \alpha < + \infty}
     \left[\alpha \wedge
         \mu\left(L \cap
          \left(
          D'_{mp}
          \right)_{\alpha}
          \right)
     \right] + \displaystyle\frac{\epsilon}{2} \\
  < & {\epsilon}.
\end{array}
$$
The proof of (1) is now completed. \hfill $\Box$

\end{slide}


%**************************




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\end{document}
