%%%%%%%%%%%%%%%%%%%%%%%%%% author.tex %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%------ NLMUA2011 Conference: paper ID 91                        %%
%%  Autocontinuity of monotone measure and convergence in measure %%
%%                    J. Li, M. Yasuda, L. Zhou                   %%
%%                                                                %%
%% ---2011.03.31 submitted to NLMUA2011                           %%
%% ---2011.05.20 final version(camera-ready)                      %%                                                                %%
%%%%%%%%%%%%%%%%%%%%%%%% Springer-Verlag %%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{document}
%
\title*{Autocontinuity from below of set functions and convergence in measure}
\author{Jun Li, \ Masami Yasuda and Ling Zhou}
\institute{Jun Li \at School of Science, Communication University of China,
                       Beijing 100024, China \\
 \email{{\bf{lijun@cuc.edu.cn}}}
 \and   Masami Yasuda \at Department of Mathematics \& Informatics, Faculty of Science,
                          Chiba University, Chiba 263-8522, Japan \\
 \email{{\bf{yasuda@math.s.chiba-u.ac.jp}}}
 \and   Ling Zhou \at College of Applied Sciences, Xinjiang Agricultural University,
                      Xinjiang, Urumqi 830052, China \\
 \email{{\bf{zhoulin5703@163.com}}}  }
%
\maketitle
%
\abstract{ In this note, the concepts of strong autocontinuity from
below and strong converse autocontinuity from below of set function
are introduced. By using four types of autocontinuity from below of
monotone measure, the relationship between convergence in measure
and pseudo-convergence in measure for sequence of measurable
function are discussed. }
%
\keywords{ Monotone measure, Autocontinuity from below, Convergence
in measure, Pseudo-convergence in measure. }
%
%----------------Introduction--------------------------
\section{Introduction}
In non-additive measure theory, there are several different kinds of
convergence for sequence of measurable functions, such as almost
everywhere convergence, pseudo-almost everywhere convergence,
convergence in measure, and convergence pseudo-in measure. The
implication relationship between such convergence concepts are
closely related to the structural characteristics of set functions.
In this direction there are a lot of results
(\cite{Li1997,Li2001,Li2003,Li2005,Liu01,LiS2005,LiM2010,
Muro2004,Pap95,Sun94,taka2005,Wang1985,Wang2009}).

In this note, we further discuss the relationship between
convergence in measure and convergence pseudo-in measure for
sequence of measurable functions. We shall introduce the concepts of
strong autocontinuity from below and strong converse autocontinuity
from below of a set function. By using the two types of
autocontinuity from below of monotone measures, we investigate the
inheriting of convergence in measure and convergence pseudo-in
measure for sequence of measurable function under the common
addition operation ``+" and logic addition operation ``$\vee$". The
implication relationship between convergence in measure and
pseudo-convergence in measure are shown by using autocontinuity from
below and converse autocontinuity from below, respectively.
%
%----------------Preliminaries--------------------------
\section{Preliminaries}
Let $X$ be a non-empty set, ${\cal F}$ a $\sigma$-algebra of subsets
of $X$, and $(X, {\cal F})$ denotes the measurable space.

\begin{definition} \ {\rm(\cite{Pap95,Wang2009})} \
  \label{df:2.1}
 Set function $\mu:{\cal F} \to [0, +\infty]$ is called a
{\it monotone measure} on $(X, {\cal F})$ iff it satisfies the
following requirements:
\begin{itemize}
 \item[{\rm (1)}] \ \ $\mu(\emptyset)=0;$
        \hfill  (vanishing at $\emptyset$)
 \item[{\rm (2)}] \ \ $A \subset B$ and $A,
   B \in {\cal F}$ \ $\Rightarrow$ \  $\mu(A) \leq \mu(B)$.
       \hfill  (monotonicity)
\end{itemize}
\end{definition}

When $\mu$ is a monotone measure, the triple $(X, {\mathcal{F}},
\mu)$ is called a monotone measure space (\cite{Pap95,Wang2009}).
%

In some literature, a set function $\mu$ satisfying the conditions
(1) and (2) of Definition \ref{df:2.1} is called a fuzzy measure or
a non-additive measure .

In this paper, all the considered sets are supposed to belong to
${\cal F}$ and $\mu$ is supposed to be a finite monotone measure,
i.e., $\mu(X) < \infty$. All concepts and symbols not defined may be
found in \cite{Pap95,Wang2009}.

%
\begin{definition} (\cite{Dobra1980})
    \label{df:2.2}
A set function $\mu:{\cal F} \to [0, +\infty)$ is said to have
 {\it pseudometric generating property} (for short {\it p.g.p}),
 if
for any $\{ E_n \} \subset {\cal F}$ and $\{ F_n \} \subset {\cal
F}$,
$$
\mu(E_n) \vee \mu(F_n) \rightarrow 0 \  \Longrightarrow \
      \mu (E_n \cup F_n) \rightarrow 0.
$$
\end{definition}
%

{\it Note}: The concept of pseudometric generated property goes back
to Dobrakov and Farkova in seventies, and this was related to
Frechet-Nikodym topology \cite{Dobra1980,Pap95}. \\


Let ${\bf F}$ be the class of all finite real-valued measurable
functions on $(X, {\mathcal{F}}, \mu)$, and let $A\in {\cal F}, f\in {\bf F}, f_n
\in {\bf F}$~($n = 1, 2, \ldots$) and $\{f_n \}$ denote a sequence
of measurable functions.
We say that {\it $\{ f_n \}$ converges in measure $\mu$ to $f$ on $A$},
and denote it by
$f_n
\begin{array}{c}
 {\mbox{\scriptsize {$\mu$}}}
  \vspace{-1.5ex}\\
  \longrightarrow
  \vspace{-1.5ex}\\
 {\mbox{\scriptsize $A$}}
\end{array}
f$,
if for any given $\sigma > 0$,
$\displaystyle{\lim_{n \to +\infty}}
\mu(\{|f_{n} - f| \geq \sigma \} \cap A) = 0$;
{\it $\{ f_n \}$ converges pseudo-in measure $\mu$ to $f$ on $A$},
and denote it by
$f_n
\begin{array}{c}
 {\mbox{\scriptsize {$p,\mu$}}}
  \vspace{-1.5ex}\\
  \longrightarrow
  \vspace{-1.5ex}\\
 {\mbox{\scriptsize $A$}}
\end{array}
f$,
if for any given $\sigma > 0$,
$\displaystyle{\lim_{n \to +\infty}}
\mu(\{|f_{n} - f|<\sigma \} \cap A) = \mu(A)$;
{\it $\{ f_n \}$ converges pseudo-in measure $\mu$ to $f$ $in$ $A$},
and denote it by $f_n \stackrel{p.\mu} {\longrightarrow} f$ in $A$,
if $f_n \stackrel{p.\mu} {\longrightarrow} f$ on $C$
for all $C \in A \cap {\cal F}$.

%----------------- Section 3 -------------------------
\section{Autocontinuity of set function}
\label{author_sec:2}

In \cite{Wang1985} Wang introduced the concepts of autocontinuity
from below and converse-autocontinuity from below of set function,
and discussed the convergence for sequence of measurable functions
by using the structure of set functions. Now we shall introduce the
concepts of strong autocontinuity from below and strong
converse-autocontinuity from below for set functions and show their
properties.
%

\begin{definition} (\cite{Pap95,Wang1985,Wang2009}) \
    \label{df:3.1}
Let $(X, {\cal F}, \mu)$ be a monotone measure space.

(1) \ $\mu$ is said to be {\it autocontinuous from below} and denote
it by {\it autoc}.$\uparrow$, if for any $E \in {\cal F}, \{F_n \}
\subset {\cal F}$,
$$
\mu(F_n) \rightarrow 0 \ \Longrightarrow \
\mu(E - F_n) \rightarrow \mu(E);
$$

(2) \ $\mu$ is said to be
{\it converse-autocontinuous from below} and denote it by {\it c.autoc}.$\uparrow$, if
for any $A \in {\cal F}, \{B_n \} \subset A\cap {\cal F}$,
$$
\mu(B_n) \to \mu(A) \ \Longrightarrow \ \mu(A - B_n) \to 0.
$$
\end{definition}
%

\begin{definition}  \
    \label{df:3.2}
Let $(X, {\cal F}, \mu)$ be a monotone measure space.

(1) \ $\mu$ is said to be {\it strong autocontinuous from below}
       and denote it by {\it s.autoc.$\uparrow$}, if
\[
\mu(E_n) \vee \mu(F_n) \rightarrow 0 \Longrightarrow
\mu (A - E_n \cup F_n) \rightarrow \mu(A),
\]
for any $A \in {\cal F}, \{ E_n \} \subset {\cal F}$ and
$\{ F_n \} \subset {\cal F}$;

(2) \ $\mu$ is said to be {\it strong converse-autocontinuous from below}
and denote it by {\it s.c.autoc.$\uparrow$}, if
\[
\mu(A - E_n) \wedge \mu(A - F_n) \rightarrow \mu(A)
\Longrightarrow
\mu(E_n \cup F_n) \rightarrow 0,
\]
for any $A \in {\cal F}$,
$\{ E_n \} \subset A \cap {\cal F}$ and
$\{ F_n \} \subset A \cap {\cal F}$.
\end{definition}
%

\begin{proposition}
 \label{Prop-31}
If $\mu$ is {\it s.autoc.$\uparrow$} (resp. {\it s.c.autoc.$\uparrow$}),
then it is {\it autoc.$\uparrow$}
(resp. {\it c.autoc.$\uparrow$}).
\end{proposition}
%
\begin{proposition}
 \label{Prop-32}
If $\mu$ is {\it autoc.$\uparrow$} and has $p.g.p$,
then it is {\it s.autoc.$\uparrow$}.
\end{proposition}
%
\begin{proposition}
 \label{Prop-33}
If $\mu$ is {\it c.autoc.$\uparrow$} and has $p.g.p$,
then it is {\it s.c.autoc.$\uparrow$}.
\end{proposition}




%------------------Section 4----------------------------
\section{Convergence in measure}
%

In this section, we study the application relationship between
convergence in measure and convergence pseudo-in measure on monotone
measure spaces. \\

The first conclusion of the following theorem due to Wang
\cite{Wang2009}.

\begin{theorem}
    \label{Th-41}
\ Let $\mu$ be a monotone measure. Then,

(1) \ $\mu$ is autoc.$\uparrow$ iff $f_{n}
\begin{array}{c}
 {\mbox{\scriptsize {$p.\mu$}}}
  \vspace{-1.5ex}\\
  \longrightarrow
  \vspace{-1.5ex}\\
 {\mbox{\scriptsize $A$}}
\end{array}
f$ whenever $f_n
\begin{array}{c}
 {\mbox{\scriptsize {$\mu$}}}
  \vspace{-1.5ex}\\
  \longrightarrow
  \vspace{-1.5ex}\\
 {\mbox{\scriptsize $A$}}
\end{array}
f$, ~$\forall A \in {\cal F}, f, f_n \in {\bf F}$;

(2) \ $\mu$ is c.autoc$\uparrow$, iff $f_{n}
\begin{array}{c}
 {\mbox{\scriptsize {$\mu$}}}
  \vspace{-1.5ex}\\
  \longrightarrow
  \vspace{-1.5ex}\\
 {\mbox{\scriptsize $A$}}
\end{array}
f$ whenever $f_n
\begin{array}{c}
 {\mbox{\scriptsize {$p.\mu$}}}
  \vspace{-1.5ex}\\
  \longrightarrow
  \vspace{-1.5ex}\\
 {\mbox{\scriptsize $A$}}
\end{array}
f$, ~$\forall A \in {\cal F}, f, f_n \in {\bf F}$.
\end{theorem}
%
\begin{proof}
\smartqed
 We only prove (2). Let $\mu$ be c.autoco$\uparrow$. If $f_n
\begin{array}{c}
 {\mbox{\scriptsize {$p.\mu$}}}
  \vspace{-1.5ex}\\
  \longrightarrow
  \vspace{-1.5ex}\\
 {\mbox{\scriptsize $A$}}
\end{array}
f$, then for any given $\sigma>0$, we have
\[
\displaystyle{\lim_{n \to +\infty}} \mu(\{|f_n - f| < \sigma \} \cap
A) = \mu(A)
\]
and therefore, using the converse-autocontinuity from below of
$\mu$, we have
%
\begin{eqnarray}
\displaystyle{\lim_{n \to +\infty}} \mu(\{|f_n - f| \geq \sigma \}
\cap A) &=&
\displaystyle{\lim_{n \to +\infty}} \mu(A - \{|f_n - f| < \sigma \}) \nonumber \\
&=& 0. \nonumber
\end{eqnarray}
%
So $f_n
\begin{array}{c}
 {\mbox{\scriptsize {$\mu$}}}
  \vspace{-1.5ex}\\
  \longrightarrow
  \vspace{-1.5ex}\\
 {\mbox{\scriptsize $A$}}
\end{array}
f$.

Conversly, for any $A \in {\cal F}, \{ B_n \} \subset A \cap {\cal
F}$, and $\mu(B_n) \to \mu(A)$, we define measurable function
sequences $\{ f_n \}$ by
$$
 f_n = {\chi}_{B_n } = \left\{
       \begin{array}{ll}
            0             & \mbox{if} \ \ x \notin B_{n} \\
            1             & \mbox{if} \ \ x  \in  B_{n},
       \end{array} \right.
$$
n = 1, 2,..., and denote $f \equiv 1$. It is easy to see that $f_n
\begin{array}{c}
 {\mbox{\scriptsize {$p.\mu$}}}
  \vspace{-1.5ex}\\
  \longrightarrow
  \vspace{-1.5ex}\\
 {\mbox{\scriptsize $A$}}
\end{array}
f$. If it implies $f_n
\begin{array}{c}
 {\mbox{\scriptsize {$\mu$}}}
  \vspace{-1.5ex}\\
  \longrightarrow
  \vspace{-1.5ex}\\
 {\mbox{\scriptsize $A$}}
\end{array}
f$, then for $\sigma = \frac{1}{2}$, we have
$$
\displaystyle{\lim_{n \to +\infty}} \mu(\{|f_{n} - f| \geq
\frac{1}{2} \} \cap A) = 0.
$$
As
$$
\{|f_{n} - f| \geq \frac{1}{2} \} \cap A = \{ 1 - {\chi}_{B_n} \geq
\frac{1}{2} \} \cap A = A - B_{n}.
$$
So $\displaystyle{\lim_{n \to +\infty}} \mu(A - B_{n}) = 0.$ This
shows that $\mu$ is c.autoc$\uparrow$. \qed
\end{proof}
%

The following theorems describe the inheriting of convergence in
measure and convergence pseudo-in measure for sequence of measurable
function under the common addition operation.
\begin{theorem}
    \label{Th-42}
\ Let $\mu$ be a monotone measure.

(1) If $\mu$ is s.autoc.$\uparrow$, then
 $f_n \stackrel{\mu} {\longrightarrow} f$ and
 $g_n \stackrel{\mu} {\longrightarrow} g$
  on $A$ imply
$$
\alpha f_n + \beta g_n
\begin{array}{c}
 {\mbox{\scriptsize {$p.\mu$}}}
  \vspace{-1.5ex}\\
  \longrightarrow
  \vspace{-1.5ex}\\
 {\mbox{\scriptsize $D$}}
\end{array}
\alpha f + \beta g,
$$
for any $D \in A \cap {\cal F}, \alpha, \beta \in R^{1}.$

(2) If $\mu$ is s.c.autoc.$\uparrow$, then
 $f_n
  \begin{array}{c}
   {\mbox{\scriptsize {$p.\mu$}}}
   \vspace{-1.5ex}\\
   \longrightarrow
   \vspace{-1.5ex}\\
   {\mbox{\scriptsize $A$}}
  \end{array}
 f$
and
 $g_n
 \begin{array}{c}
  {\mbox{\scriptsize {$p.\mu$}}}
  \vspace{-1.5ex}\\
  \longrightarrow
  \vspace{-1.5ex}\\
  {\mbox{\scriptsize $A$}}
 \end{array}
 g$
imply
 $$
 \alpha f_n + \beta g_n
 \begin{array}{c}
 {\mbox{\scriptsize {$\mu$}}}
  \vspace{-1.5ex}\\
  \longrightarrow
  \vspace{-1.5ex}\\
 {\mbox{\scriptsize $A$}}
 \end{array}
 \alpha f + \beta g,
 $$
for any $A \in {\cal F}$, $\alpha, \beta \in R^{1}$.
\end{theorem}
\begin{proof}
\smartqed It is similar to the proof of Theorem~\ref{Th-41}. \qed
\end{proof}

The following Theorem~\ref{Th-43} and \ref{Th-44} describe
respectively the characteristics of strong autocontinuity from below
and strong converse-autocontinuity from below of set functions.

\begin{theorem} \
   \label{Th-43}
The following statements are equivalent:

(1)$\quad$$\mu$ is s.autoc.$\uparrow$;

(2)$\quad$$f_{n}+g_{n}
\begin{array}{c}
 {\mbox{\scriptsize {$p.\mu$}}}
  \vspace{-1.5ex}\\
  \longrightarrow
  \vspace{-1.5ex}\\
 {\mbox{\scriptsize $A$}}
\end{array}
0$ whenever $f_n
\begin{array}{c}
 {\mbox{\scriptsize {$\mu$}}}
  \vspace{-1.5ex}\\
  \longrightarrow
  \vspace{-1.5ex}\\
 {\mbox{\scriptsize $A$}}
\end{array}
0$ and $g_n
\begin{array}{c}
 {\mbox{\scriptsize {$\mu$}}}
  \vspace{-1.5ex}\\
  \longrightarrow
  \vspace{-1.5ex}\\
 {\mbox{\scriptsize $A$}}
\end{array}
0$, $\forall A \in {\cal F}$;

(3)$\quad$$f_{n} \vee g_{n}
\begin{array}{c}
 {\mbox{\scriptsize {$p.\mu$}}}
  \vspace{-1.5ex}\\
  \longrightarrow
  \vspace{-1.5ex}\\
 {\mbox{\scriptsize $A$}}
\end{array}
0$ whenever $f_n
\begin{array}{c}
 {\mbox{\scriptsize {$\mu$}}}
  \vspace{-1.5ex}\\
  \longrightarrow
  \vspace{-1.5ex}\\
 {\mbox{\scriptsize $A$}}
\end{array}
0$ and $g_n
\begin{array}{c}
 {\mbox{\scriptsize {$\mu$}}}
  \vspace{-1.5ex}\\
  \longrightarrow
  \vspace{-1.5ex}\\
 {\mbox{\scriptsize $A$}}
\end{array}
0$, $\forall A \in {\cal F}$.
\end{theorem}
\begin{proof}
\smartqed
 (1) $\Longrightarrow$ (2). It follows directly from Theorem~\ref{Th-42}
above.

(2) $\Longrightarrow$ (3). For any $A \in {\cal F}$, if $f_n
\begin{array}{c}
 {\mbox{\scriptsize {$\mu$}}}
  \vspace{-1.5ex}\\
  \longrightarrow
  \vspace{-1.5ex}\\
 {\mbox{\scriptsize $A$}}
\end{array}
0$ and $g_n
\begin{array}{c}
 {\mbox{\scriptsize {$\mu$}}}
  \vspace{-1.5ex}\\
  \longrightarrow
  \vspace{-1.5ex}\\
 {\mbox{\scriptsize $A$}}
\end{array}
0$, then
$|f_n|
 \begin{array}{c}
 {\mbox{\scriptsize {$\mu$}}}
  \vspace{-1.5ex}\\
  \longrightarrow
  \vspace{-1.5ex}\\
 {\mbox{\scriptsize $A$}}
 \end{array}
0$ and $|g_n|
\begin{array}{c}
 {\mbox{\scriptsize {$\mu$}}}
  \vspace{-1.5ex}\\
  \longrightarrow
  \vspace{-1.5ex}\\
 {\mbox{\scriptsize $A$}}
\end{array}
0$.
By condition (2), we have $|f_{n}|+|g_{n}| \stackrel{p.\mu}
{\longrightarrow}0$ on $A$, therefore, for any $\sigma >0$,
$$
\displaystyle{\lim_{n \to +\infty}} \mu(\{|f_{n}|+|g_{n}|<\sigma \}
\cap A) = \mu(A).
$$
Noting that $|f_{n} \vee g_{n}| \leq |f_{n}|+|g_{n}|$, we get
$$
\{|f_{n}|+|g_{n}|<\sigma \} \cap A \subseteq \{|f_{n} \vee
g_{n}|<\sigma \} \cap A \subseteq A.
$$
So
$$
\displaystyle{\lim_{n \to +\infty}} \mu(\{|f_{n} \vee g_{n}|<\sigma
\} \cap A) = \mu(A).
$$
This shows $f_{n} \vee g_{n} \stackrel{p.\mu} {\longrightarrow}0$ on
$A$.

(3)$\Longrightarrow$(1). For any $\{ E_n \} \subset {\cal F}, \{ F_n
\} \subset {\cal F}$ with $\displaystyle{\lim_{n \to \infty }}
\mu(E_n) \vee \mu(F_n) = 0$, we define measurable function sequences
$\{ f_n \}$ and $\{ g_n \}$ by
$$
 f_n = {\chi}_{E_n} = \left\{
       \begin{array}{ll}
            0             & \mbox{if} \ \ x \notin E_n\\
            1             & \mbox{if} \ \ x  \in  E_n
       \end{array} \right.
$$
and
$$
 g_n = {\chi}_{F_n} = \left\{
       \begin{array}{ll}
            0             & \mbox{if} \ \ x \notin F_n\\
            1             & \mbox{if} \ \ x  \in  F_n,
       \end{array} \right.
$$
n = 1, 2,..., then $f_n \stackrel{\mu} {\longrightarrow} 0$ on $A$
and $g_n \stackrel{\mu} {\longrightarrow} 0$ on $A$. Thus, $f_n \vee
g_n \stackrel{\mu} {\longrightarrow} 0$ on $A$. Therefore for
$\sigma = \frac{1}{2}$, we have
$$
\displaystyle{\lim_{n \to +\infty}} \mu(\{f_{n} \vee g_{n}
<\frac{1}{2} \} \cap A) = \mu(A).
$$
Noting $f_{n} \vee g_{n} = {\chi}_{E_n} \vee {\chi}_{F_n} =
{\chi}_{E_n \cup F_n}$, and
$$
\{ {\chi}_{E_n} \vee {\chi}_{F_n} < \frac{1}{2} \} \cap A = A - \{
{\chi}_{E_n} \vee {\chi}_{F_n} \geq \frac{1}{2} \} = A - E_{n} \cup
F_{n}.
$$
So
$$
\displaystyle{\lim_{n \to +\infty}} \mu(A - E_{n} \cup F_{n}) =
\mu(A).
$$
That is, $\mu$ is s.autoc.$\uparrow$. \qed
\end{proof}

\begin{theorem} \
      \label{Th-44}
The following statements are equivalent:

(1)$\quad$$\mu$ is s.c.autoc.$\uparrow$;

(2)$\quad$$f_{n}+g_{n}
\begin{array}{c}
 {\mbox{\scriptsize {$\mu$}}}
  \vspace{-1.5ex}\\
  \longrightarrow
  \vspace{-1.5ex}\\
 {\mbox{\scriptsize $A$}}
\end{array}
0$ whenever $f_n
\begin{array}{c}
 {\mbox{\scriptsize {$p.\mu$}}}
  \vspace{-1.5ex}\\
  \longrightarrow
  \vspace{-1.5ex}\\
 {\mbox{\scriptsize $A$}}
\end{array}
0$ and $g_n
\begin{array}{c}
 {\mbox{\scriptsize {$p.\mu$}}}
  \vspace{-1.5ex}\\
  \longrightarrow
  \vspace{-1.5ex}\\
 {\mbox{\scriptsize $A$}}
\end{array}
0$, $\forall A \in {\cal F}$;

(3)$\quad$$f_{n} \vee g_{n}
\begin{array}{c}
 {\mbox{\scriptsize {$\mu$}}}
  \vspace{-1.5ex}\\
  \longrightarrow
  \vspace{-1.5ex}\\
 {\mbox{\scriptsize $A$}}
\end{array}
0$ whenever $f_n
\begin{array}{c}
 {\mbox{\scriptsize {$p.\mu$}}}
  \vspace{-1.5ex}\\
  \longrightarrow
  \vspace{-1.5ex}\\
 {\mbox{\scriptsize $A$}}
\end{array}
0$ and $g_n
\begin{array}{c}
 {\mbox{\scriptsize {$p.\mu$}}}
  \vspace{-1.5ex}\\
  \longrightarrow
  \vspace{-1.5ex}\\
 {\mbox{\scriptsize $A$}}
\end{array}
0$, $\forall A \in {\cal F}$.
\end{theorem}

\begin{proof}
\smartqed
It is similar to the proof of Theorem~\ref{Th-43}.  \qed
\end{proof}
%
\begin{acknowledgement}
The first author was supported by NSFC Grant No.70771010. The second
author was supported by JSPS Grant No.22540112.
\end{acknowledgement}
%


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%
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\end{thebibliography}

\end{document}

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