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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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%%--2006/03/12(Li)                                         %%
%%--revised version--2006/03/22---from Li Jun    
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%%-- ipmu2006_lly_C.tex  --2006/03/22---(Yasuda)
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%%-- RIMS2006    
%%   ipmu2006_lly_C2.tex  --2006/09/22---(Yasuda)
%%   零加法性Lusin定理によるFuzzy neural network の近似に
%%   ついて
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\documentclass{ipmu2006}
\usepackage{latexsym,amssymb,theorem}
\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]
%---------------------Text-------------------------------
\title{\bf 
零加法性Lusin定理による Fuzzy neural networkの近似について
(Approximation of fuzzy neural networks by 
             using Lusin's theorem) }
\author{
{\bf Jun Li\thanks{
           Supported by the postdoctoral research
             finance of Communication University of China.} }\\
  School of Sciences,\\
  Commun.~Univ.~of China, \\
  Beijing 100024, China\\
  lijun@cuc.edu.cn
\And
{\bf Jianzeng Li}\\
  School of Information Engi., \\
  Commun.~Univ.~of China, \\
  Beijing 100024, China\\
  jzli@cuc.edu.cn
\And
{\bf Masami Yasuda}\\
  Faculty of Science, \\
  Chiba University, \\
  Chiba 263-8522, Japan\\
  yasuda@math.s.chiba-u.ac.jp}
%
%
\begin{document}
\maketitle
\begin{abstract}
Regular fuzzy neural network (RFNN) に関して、ファジィ積分による
近似法を提案した。任意のファジィ値可測関数に対して、
4層RFNNは有限な$\mathbb{R}$上の劣加法性測度によるファジィー積分
ノルムの意味で近似できる。

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%       Section 1           %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%\hspace{4mm} 
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. The studies on this
matter were undertaken by many authors and a great number of
important results were obtained (\cite{Atta97,Burt98,Scar98} etc).
The similar approximation problems in fuzzy environment were
investigated by Buckley \cite{Buck93,Buck94}, P. Liu
\cite{Liup00,Liup01} and other authors. In \cite{Liup01} Liu
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 $\mathbb{R}$.
%


%********************************
\section{Preliminaries}
% section 2
%\hspace{4mm} 
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 ${\cal O}$.

%\hspace{4mm} 
A set function $\mu:{\cal B} \to [0, +\infty]$ is
called
 {\it a fuzzy measure}(\cite{Naru04}),
 if it satisfies the following properties:
(FM1)\ $\mu(\emptyset)=0$; (FM2)\ $A \subset B$ \ implies \
$\mu(A) \leq \mu(B)$. A fuzzy measure $\mu$ is called
 {\it null-additive} (\cite{Wang92}), if for any
$E, F \in {\cal B}$ and $\mu(F) = 0$ imply
 $\mu(E \cup F) = \mu(E)$;
 {\it sub-additive} (\cite{Pap95}), if
for any $E, F \in {\cal B}$ we have
 $\mu (E \cup F) \leq \mu(E)+\mu(F)$.
%

%\hspace{4mm} 
In this paper, we always assume that $\mu$ is a
finite, sub-additive and continuous fuzzy measure on ${\cal B}$.

%\hspace{4mm} 
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
\[
  \begin{array}{ll}
    & (S)\displaystyle\int_{A} f \, d\mu \\[2ex]
   \triangleq  & \displaystyle{\bigvee_{0 \leq \alpha <
   + \infty}}
   \left[ \alpha \wedge \mu(\{ x : f(x) \geq \alpha \}
     \cap A) \right]
  \end{array}
\]
%
\begin{thm}{\rm(Lusin's theorem cf. \cite{Li04b,song03})}
 \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}
%
%*****************************************
\section{Approximation in fuzzy mean by regular fuzzy neural 
networks}
% section 3
%\hspace{4mm} 
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 
$\mathbb{R}$.

%\hspace{4mm} 
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
$$
  |\tilde{A}|
      \triangleq
      \bigvee_{\alpha\in[0, 1]}(|a_{\alpha}^{-}|\vee|a_{\alpha}^{+}|).
$$

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 $A, B \subset
\mathbb{R}$,
\[
  \begin{array}{ll}
             &  d_{H}(A, B) \\[2ex]
   \triangleq \! \!  &  \! \! \max \left\{
            \displaystyle\sup_{x\in A}\inf_{y\in B}(|x-y|),%\right. \\[2ex]
             %&  \qquad \, \, \left.
             \displaystyle\sup_{y\in B}\inf_{x\in A}(|x-y|) \right\} .
 \end{array}
\]

%\hspace{4mm} 
It is known that $({\mathcal{F}}_{0}(\mathbb{R}), d)$
is a completely separable metric space (\cite{Diam94}).
Also we note that the next assertion which is used in 
later.

\begin{prop}\label{Prop-00} {\rm(\cite{Liup01})}
%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}})$,
  \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}

By the well known extension principle, each function 
$f: \mathbb{R}^n \rightarrow \mathbb{R} $ may be extended to
one $ {\mathcal{F}}_{0}(\mathbb{R})^n \rightarrow 
{\mathcal{F}}(\mathbb{R}) $ and, for each fuzzy number,
the addition, the multiplication and the multiplication by
a scalar are defined by the extension princile
{\rm(\cite{Liup01})}.

%\hspace{4mm} 
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
 \[
 \tilde{F}: T \rightarrow
{\mathcal{F}}_{0}(\mathbb{R}).
\]

%\hspace{4mm} 
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
\[
{\bigtriangleup}_{S}(\tilde{F_{1}}, \tilde{F_{2}})
 \triangleq (S) \int_{T} d(\tilde{F_{1}}, \tilde{F_{2}}) d\mu.
\]
%
\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}
{\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*}
Because of the elementary inequality:
$
  a \wedge(b+c) \leq (a\wedge b) + (a \wedge c)
$
where $a,b,c \geq 0$, we have
\begin{eqnarray*}
  & &{\bigtriangleup}_{S}(\tilde{F_{1}}, \tilde{F_{3}}) \\[2mm]
  &\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]
  &\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]
  &\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*}
%
\begin{df}{\rm (\cite{Liup01}) \
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}

%\hspace{4mm} 
Immediately, if $\mathcal{S}(T)$ denotes the set of
all fuzzy-valued simple functions, then $\mathcal{S}(T) \subset
\mathcal{L}(T)$.
%

%\hspace{4mm} 
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}
%
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
$x \in \mathbb{R}, \tilde{W}_{i},
\tilde{V}_{ij}, \tilde{U}_{j}, \tilde{\Theta}_{j}
\in {\mathcal{F}}_{0}(\mathbb{R})$.

%\hspace{4mm} 
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. \cite{Liup01}).

%\hspace{4mm} 
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
\[
      v_i[\sigma] \triangleq
          \sum_{j=1}^{m}v_{ij} \cdot
                \sigma(x\cdot u_{j} + \theta_{j}).
\]
%

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}
%
%\hspace{4mm} 
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.
%
\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,

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

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

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 ${\bigtriangleup}_{S}(\tilde{H},
\tilde{\Phi}) < \epsilon$.
%
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*}
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*}
where the notation of set $C_{mp}$ 
is assigned as 
$$
  C_{mp} = 
          \sum_{k=1}^{m}|\tilde{A}_{k}|\cdot
          d\left(\sum_{k=1}^{p}B_{kj}, 
          \chi_{T_{k}}(x)\right).
$$
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*}
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
$$
   D'_{mp} = \sum_{k=1}^{m}|\tilde{A}_{k}|\cdot
             d\left(
             \sum_{j=1}^{p_{k}}B_{kj}^{'} -     
             \chi_{T_{k}}(x)\right).
$$ 

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 $x\in L$,
\[
  D'_{mp} = \sum_{k=1}^{m}|\tilde{A}_{k}|\cdot
          d\left(\sum_{j=1}^{p_{k}}B_{kj}^{'} 
   - \chi_{T_{k}}(x)\right)
   < \frac{\epsilon}{2}.
\]
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*}
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 $\Box$

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




%
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\end{document}
