A random version of Mazur's lemma
José M. Zapata \thanks{Universidad de Murcia
Dpto. Matemáticas
30100 Espinardo
Murcia
Spain
e-mail: jmzg1@um.es
The purpose of this paper is to generalize the classical Mazur's lemma from the classical convex analysis to the framework of locally $L^0$-convex modules. In this version an extra condition of countable concatenation is included. We provide a counterexample showing that this condition cannot be removed.
1409.7932.tex
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\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\title{A random version of Mazur's lemma}


\author{José M. Zapata \thanks{Universidad de Murcia, Dpto. Matemáticas, 30100 Espinardo, Murcia, Spain, e-mail: jmzg1@um.es}}


\date{\today}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\emph{ Draft for discussion with J. Orihuela}


\begin{abstract}
The purpose of this paper is to generalize the classical Mazur's lemma from the classical convex analysis to the framework of locally $L^0$-convex modules. In this version an extra condition of countable concatenation is included. We provide a counterexample showing that this condition cannot be removed. 
\end{abstract}

\keywords{lemma's Mazur, $L^0$-modules, locally $L^0$-convex modules, gauge function, countable concatenation, countable concatenation closure.}
 
%\tableofcontents

\section*{Introduction}

%In \cite{key-1}, motivated by the financial applications, Filipovic, Kupper and Vogelpoth try to provide an appropriate theoretical framework in order to study the conditional risk measures and develop the classical convex analysis for topological $L^0$-modules.
%
%To this end, they introduce the gauge function for $L^0$-modules and, in the same way as in the convex analysis, they claim that a topological $L^0$-module is locally $L^0$-convex if, and only if, its topology is induced by a family of $L^0$-seminorms. Namely, theorem 2.4 of \cite{key-1}
%
%Nevertheless, in \cite{key-2} T. Guo, S. Zhao and X. Zeng warn that there is a hole in the argument and introduce some theoretical considerations. They even conjecture that theorem 2.4 of \cite{key-1} does not hold but they do not delve into the subject and no counterexample is given, focusing the rest of their analysis only over locally $L^0$-convex topologies induced by a family of $L^0$-seminorms.  
%
%In this paper, we go further and provide a characterization of locally $L^0$-convex modules induced by a family of $L^0$-seminorms. 
%Finally, taking advantage of these ideas, we will give a counterexample of a locally $L^0$-convex topology that cannot be induced by any family of $L^0$-seminorms.

In recents years, works like \cite{key-2,key-3,key-5,key-6,key-8,key-9} have highlighted that the appropriate theoretical framework in which embed the theory of conditional risk measures is the theory of locally $L^0$-convex modules.

As \cite{key-3} propose, to carry out this study it is necessary to bring tools from classical convex analysis fitting them to this new framework. Ultimately, to create a randomized generalization of convex analysis. 

Due to difficulties deriving from working with scalars into the ring $L^0$ instead of $\R$, some obstacles must be overcome, namely, as shown in \cite{key-3} and \cite{key-8}, mainly the fact that not all the non-zero elements possess a multiplicative inverse, i. e, $L^0$ is not a field, and that $L^0$ is not endowed with a total order.

Thus, some theorems from convex analysis will remain valid in the $L^0$-convex modules, but others will require additional conditions. So in \cite{key-3} and \cite{key-8}, some notions known as countable concatenation properties are addressed.

 

An important tool of classical convex analysis is the Mazur's lemma, which shows that for any weakly convergent sequence in a Banach space there is a sequence of convex combinations of its members that converges strongly to the same limit. This allows in some situations changing weak topology by strong topology and vice versa working with normed spaces.   
 

Then, the purpose of this article is to show a randomize version for $L^0$-normed modules, finding that an extracondition of countable concatenation is needed.  

This paper is structured as follow:  We give a first section of preliminaries. In the second section we study some properties of the gauge function for $L^0$-modules. And finally, the third section is devoted to the Mazur's lemma for $L^0$-modules, proving the main result and a  counterexample showing that the extra condition of countable concatenation cannot be removed.
 
\section{Preliminaries}

Given a probability space $\left(\Omega,\mathcal{F},\PP\right)$, which will be fixed for the rest of this paper, we consider $L^{0} \left(\Omega,\mathcal{F},\PP\right)$, the set of equivalence classes of  real valued $\mathcal{F}$-measurable random variables, which will be denoted simply as $L^{0}$.

It is known that the triple $\left(L^{0},+,\cdot\right)$ endowed with the partial order of the almost sure dominance is a lattice ordered ring.

We write ``$X\geq Y$`` if $\PP\left( X\geq Y \right)=1$.
Likewise, we write ``$X>Y$'', if $\PP\left( X> Y \right)=1$. 

And, given $A\in \mathcal{F}$, we write $X>Y$ (respectively,  $X \geq Y$) on $A$, if $\PP\left(X>Y \mid A\right)=1$ (respectively , if $\PP\left(X \geq Y \mid  A \right)=1$). 


We also define 

\[
L_{+}^{0}:=\left\{ Y\in L^{0};\: Y\geq 0\right\} 
\]
and
\[
L_{++}^{0}:=\left\{ Y\in L^{0};\: Y>0  \right\}. 
\]

And denote by $\bar{L^{0}}$, the set of equivalence classes of  $\mathcal{F}$-measurable random variables taking values in $\bar{\R}=\R\cup\{\pm\infty\}$, and extend the partial order of the almost sure dominance to $\bar{L^{0}}$. 


In A.5 of \cite{key-4} is proved the proposition below:

\begin{prop}
\label{prop: esssup}
Let $\phi$  be a subset of $L^{0}$, then
\begin{enumerate}
\item There exists $Y^{*}\in\bar{L^{0}}$ such that $Y^{*}\geq Y$ for all 
$Y\in\phi,$ and such that any other $Y'$ satisfying the same, verifies $Y'\geq Y^{*}.$
\item Suppose that  $\phi$ is directed upwards. Then there exists a increasing sequence  $Y_{1}\leq Y_{2}\leq...$ in $\phi,$ such that 
 $Y_{n}$ converges to $Y^{*}$ almost surely.
\end{enumerate}
\end{prop}.

\begin{defn}
Under the conditions of the previous proposition, 
$Y^{*}$ is called essential supremum of $\phi$, and we write

\[
\esssup \phi=\underset{Y\in\phi}{\esssup Y}:=Y^{*}.
\]


The essential infimum of $\phi$ is defined as

\[
\essinf \phi=\underset{Y\in\phi}{\essinf\, Y}:=\underset{Y\in\phi}{-\esssup \left(-Y\right)}.
\]
\end{defn}

%\begin{defn}
%Given a sequence $\left\{ Y_{n}\right\}_{n\in\N}$ in
%$L^{0}$, we define the essential  limit inferior (respectively, the essential  limit superior) as 
%\[
%\underset{n\rightarrow\infty}{\essliminf} Y_{n}:=\underset{n}{\esssup}\underset{m\geq n}{\essinf} Y_{m}
%\]
%(respectively, $\underset{n\rightarrow\infty}{\esslimsup} Y_{n}:=\underset{n}{\essinf}\underset{m\geq n}{\esssup} Y_{m}$).
%\end{defn}


The order of the almost sure dominance also lets us define a topology on $L^{0}$. Let 

\[
B_{\varepsilon}:=\left\{ Y\in L^{0};\:\left|Y\right|\leq\varepsilon\right\} 
\]

the ball of radius $\varepsilon\in L_{++}^{0}$ centered at $0\in L^{0}$.
Then, for all $Y\in L^{0}$, $\mathcal{\mathcal{U}}_{Y}:=\left\{ Y+B_{\varepsilon};\:\varepsilon\in L_{++}^{0}\right\} $ is a neighborhood base of $Y$. Thus, it can be defined a topology on $L^{0}$ that it will be known as the topology induced by $\left|\cdot\right|$ and $L^{0}$ endowed with this topology will be denoted by $L^{0}\left[\left|\cdot\right|\right]$. 


Now, we will give the central concepts of the theory of locally $L^0$-convex modules.

\begin{defn}
A topological $L^{0}$-module $E\left[\tau\right]$ is a $L^{0}$-module $E$ endowed with a topology $\tau$ such that 
\begin{enumerate}
\item $E\left[\tau\right]\times E\left[\tau\right]\longrightarrow E\left[\tau\right],\left(X,X'\right)\mapsto X+X'$
and
\item $L^{0}\left[\left|\cdot\right|\right]\times E\left[\tau\right]\longrightarrow E\left[\tau\right],\left(Y,X\right)\mapsto {Y}X$
\end{enumerate}
are continuous with the corresponding product topologies.
\end{defn}

\begin{defn}
A topology $\tau$ on a $L^{0}$-module $E$ is locally $L^{0}$-convex  
 if there is a neighborhood base of $0\in{E}$ $\mathcal{U}$ such that each $U\in\mathcal{U}$ is
\begin{enumerate}
\item $L^{0}$-convex, i.e. ${Y}X_{1}+{(1-Y)}X_{2}\in U$ for all $X_{1},X_{2}\in U$
and $Y\in L^{0}$ with $0\leq Y\leq1,$
\item $L^{0}$-absorbent, i.e. for all $X\in E$ there is a $Y\in L_{++}^{0}$
such that $X\in {Y}U,$
\item $L^{0}$-balanced, i.e. ${Y}X\in U$ for all $X\in U$ and $Y\in L^{0}$
with $\left|Y\right|\leq1.$

And, as discussed in \cite{key-13}, we add an extracondition

\item $U$ has the relative countable concatenation property.

In this case, $E\left[\tau\right]$ is called locally $L^0$-convex module. 
\end{enumerate}
\end{defn}

The notion of having the relative countable concatenation property will be recalled later (see Definition \ref{def: cc}).

\begin{defn}
A function $\left\Vert \cdot\right\Vert :E\rightarrow L_{+}^{0}$
is a $L^{0}$-seminorm on $E$ if:
\begin{enumerate}
\item $\left\Vert {Y}X\right\Vert =\left|Y\right|\left\Vert X\right\Vert $
for all $Y\in L^{0}$ y $X\in E.$
\item $\left\Vert X_{1}+X_{2}\right\Vert \leq\left\Vert X_{1}\right\Vert +\left\Vert X_{2}\right\Vert ,$
for all $X_{1},X_{2}\in E.$


If, moreover

\item $\left\Vert X\right\Vert =0$ implies $X=0,$
\end{enumerate}
then $\left\Vert \cdot\right\Vert $ is a $L^{0}$-norm on
$E$.

\end{defn}
\begin{defn}
Let $\mathcal{P}$ be a family of $L^0$-seminorms on a $L^{0}$-module
$E$. Suppose that $Q\subset\mathcal{P}$ is finite and $\varepsilon\in L_{++}^{0},$
we define 

\[
U_{Q,\varepsilon}:=\left\{ X\in E;\:\underset{\left\Vert .\right\Vert \in Q}{\sup}\left\Vert X\right\Vert \leq\varepsilon\right\} .
\]
Then for all $X\in E$, $\mathcal{\mathcal{U}}_{Q,X}:=\left\{ X+U_{\varepsilon};\:\varepsilon\in L_{++}^{0},\: Q\subset \mathcal{P}\: {finite} \right\} $
is a neighbourhood base of $X$. Thereby, we define a topology
on $E$, which it will be known as the topology induced by $\mathcal{P}$
and $E$ endowed with this topology will be denoted by $E\left[\mathcal{P}\right]$.


In addition, it is proved by the lemma 2.16 of \cite{key-3} that $E\left[\mathcal{P}\right]$ is a locally $L^0$-convex module.

Furthermore, according to \cite{key-13} a topological $L^0$-module $E\left[\tau\right]$ is a locally $L^0$-convex module if, and only if, $\tau$ is induced by a family of $L^0$-seminorms.
\end{defn}

\begin{defn}
Given a topological $L^{0}$-module $E\left[\tau\right]$, we denote by $E[\tau]^*$, or simply by $E^*$, the $L^0$-module of continuous $L^0$-linear functions $\mu : E \rightarrow L^0$.

We define
\[
\begin{array}{c}
\left\langle \cdot, \cdot\right\rangle : E \times E^*\longrightarrow L^0\\
\left\langle X, X^* \right\rangle:=X^*(X).
\end{array}
\]

For each $X^* \in E^*$ it holds that 
\[
\begin{array}{c}
q_{X^*}:E\rightarrow L^0_+ \\
q_{X^*}(X):=\left|\left\langle X, X^* \right\rangle\right|
\end{array}
\]
 is a $L^0$-seminorm. 

Now, consider the topology $\sigma(E,E^*)$ induced by the family of $L^0$-seminorms
\[
\left\{q_{X^*} ; \: X^* \in E^* \right\}.
\]
Then, $\sigma(E,E^*)$ is a locally $L^0$-convex topology, which is called the weak topology of $E$. 


Likewise, for each $X\in E$ it holds that
\[
\begin{array}{c}
q_{X}:E^*\rightarrow L^0_+ \\
q_{X}(X^*):=\left|\left\langle X, X^* \right\rangle\right|
\end{array}
\]
 is a $L^0$-seminorm.

And we have the  $L^0$-convex topology $\sigma(E^*,E)$ induced by the family of $L^0$-seminorms
\[
\left\{q_{X} ; \: X \in E \right\},
\]
which is called the weak-$*$ topology of $E$.  
\end{defn}
\section{The gauge function and the countable concatenation closure.}


Let us write the notion of gauge function introduced in \cite{key-3}:  

\begin{defn}
Let $E$ be a $L^0$-module. The gauge function $p_{K}:E\rightarrow\bar{L}_{+}^{0}$ of
a set $K\subset E$ is defined by 

\[
p_{K}\left(X\right):=\essinf\left\{ Y\in L_{+}^{0};\: X\in {Y}K\right\} .
\]

In addition,  if $K$ is $L^{0}$-convex, $L^{0}$-absorbent and $L^0$-balanced, then $p_{K}$ is a $L^0$-seminorm (see \cite{key-3}, Proposition 2.23).

\end{defn}



%We also have the next result (see 2.4 of \cite{key-1} and 2.22 of \cite{key-2}):
%
%\begin{prop}
%The gauge function $p_{U}$ of a $L^{0}$-convex and $L^{0}$-absorbent set $U\subset$ satisfies: 
%\begin{enumerate}
%\item $p_{U}(X)\geq1$ on $B$ for all $X\in E$ with $1_{A}X\notin1_{A}U$
%for all $A\in\mathcal{F}$ with $P(A)>0$, $A\subset B$.
%\item If in addition, $E$ is a locally $L^{0}$-convex module, then
%\[
%\overset{0}{U}\subset\left\{ X\in E;\: p_{U}(X)<1\right\} 
%\]
%\end{enumerate}
%\end{prop}

%Proceeding in the same way as the classical convex analysis, given a $L^{0}$-convex, $L^{0}$-absorbent and $L^{0}$-balanced set $U\subset E$, one can expect that $\left\{ X\in E;\: p_{U}(X)<1\right\} \subset U$ holds. If this held, we could prove that any topological $L^0$-module is locally $L^0$-convex module if, and only if, its topology is induced by a family of $L^0$-seminorms.
%
%Not in vain, this statement is set as valid in theorem 2.4 given in \cite{key-1}.
%
%
%However, in \cite{key-3} the authors point out that the argument that leads to theorem 2.4 given in \cite{key-1} has a hole and conjecture that, according to their observations, in  general the topology of a locally $L^0$-convex module is not necessary induced by a family of $L^0$-seminorms, but no counterexample is given, and they focus the rest of their analysis only over locally $L^0$-convex topologies induced by a family of $L^0$-seminorms. This is so because the spaces usually used for financial applications are always endowed with a family of $L^0$-seminorms.    
%
%
%We go a little bit further in this subject and provide an example (see \ref{contraejemplo}) of a locally $L^0$-convex topology not induced by any family of $L^0$-seminorms. Therefore, the theorem 2.4 given in \cite{key-1} does not hold.
%
%
%
%In addition, this example shows that there exists a $L^{0}$-convex, $L^{0}$-absorbent and $L^{0}$-balanced set $U\subset E$ such that $\left\{ X\in E;\: p_{U}(X)<1\right\} \nsubseteq U$.
%~\\

%Let us introduce some notation:

%Given a $L^0$-module $E$, we denote by $\Pi(\Omega,\mathcal{F})$ the set of countable partitions on $\Omega$ to $\mathcal{F}$. 
%\[
%\Pi\left(\Omega,\mathcal{F}\right):=\left\{ \left\{ A_{n}\right\} _{n\in I};\begin{array}{c}
%I\subset\mathbb{N}\\
%A_{i}\cap A_{j}=\phi\textnormal{ for all }i\neq j\in I\\
%\Omega=\bigcup_{n\in I}A_{n}\\
%A_{n}\in\mathcal{F}\textnormal{ for all }n\in I
%\end{array}\right\}
%\]

%T. Guo introduce a notion in \cite{key-2}, in order to be able to define the formal sum $\underset{n\in\mathbb{N}}{\sum}1_{A_{n}}X_{n}$
%where $X_{n}\in E$ and $\left\{ A_{n}\right\} _{n\in\mathbb{N}}\in\Pi\left(\Omega,\mathcal{F}\right)$. 
%\begin{defn}

%\end{defn}
%Let $E$ be a $L^{0}$-module. Given a set $C\subset E$, we call the countable concatenation closure of $C$ the set 

%%%%%%%%%%%%%%%%%%%%

Now we will give the notion of having the relative countable concatenation property, which is based on the notion of countable concatenation property given in \cite{key-8}. In \cite{key-3} the authors work with two others notions of countable concatenation property, one for the topology and other for the family of $L^0$-seminorms, although both properties turn out to be the same.
 
The notion introduced in \cite{key-8}, and the one given below, are related to the $L^0$-module itself rather than the topology.  

\begin{defn}
\label{def: cc}
Let $E$ be a $L^{0}$-module and $K\subset E$ a subset, and denote by $\Pi\left(\Omega,\mathcal{F}\right)$ the set of countable partitions on $\Omega$ to $\mathcal{F}$.
\begin{itemize}
\item  Given a sequence $\left\{ X_{n}\right\}_{n\in \mathbb{N}}$
in $E$ and a partition $\left\{ A_{n}\right\}_{n\in \mathbb{N}}\in\Pi(\Omega,\mathcal{F})$, 
we define the set of countable concatenations of $\{X_n\}_n$ and $\{A_n\}_n$ as
\[
cc\left(\{A_{n}\}_n,\{X_{n}\}_n\right):=\{X\in E;\: 1_{A_{n}}X_{n}=1_{A_{n}}X \textnormal{ for each } n\in \mathbb{N}\}.
\]

 %\item  We say that $K$ has \emph{the countable concatenation property}, if for each sequence $\left\{ X_{n}\right\}_{n\in\mathbb{N}}$
%in $K$ and each partition 
%$\left\{ A_{n}\right\}_{n\in\mathbb{N}}\in\Pi(\Omega,\mathcal{F})$ it holds
%\[
%cc\left(\{A_{n}\}_n,\{X_{n}\}_n\right) \textnormal{is non empty, and } cc\left(\{A_{n}\}_n,\{X_{n}\}_n\right) \subset K.
%\]

\item   $K$ is said to have the \emph{relative countable concatenation property}, if for each sequence $\left\{ X_{n}\right\}_{n}$
 in $K$ and each partition
$\left\{ A_{n}\right\}_{n}\in\Pi(\Omega,\mathcal{F})$ 
it holds  
\[
cc\left(\{A_{n}\}_n,\{X_{n}\}_n\right)\subset K.
\]


   
\item  We call the countable concatenation closure of $K$ the set defined below 
\[
\overline{K}^{cc}:=\bigcup cc\left(\{A_{n}\}_n,\{X_{n}\}_n\right)
\]
%\[
%=\{X\in E; \: \exists \{A_n\}_{n\in\N} \in \Pi(\Omega,\mathcal{F}) \textnormal{ with } 1_{A_n}X\in 1_{A_n}C  \} 
%\]

where $\left\{ A_{n}\right\}_{n}$ runs through $\Pi(\Omega,\mathcal{F})$ and $\left\{ X_{n}\right\}_{n}$
runs though the sequences in $K$. Or in another way written
\[
\overline{K}^{cc}=\{X\in E; \: \exists \{A_n\}_{n\in\N} \in \Pi(\Omega,\mathcal{F}) \textnormal{ with } 1_{A_n}X\in 1_{A_n}K \textnormal{ for all }n  \}. 
\]
\end{itemize}
\end{defn}

It is clear that $K$ has the relative countable concatenation property if, and only if, $\overline{K}^{cc}=K$.


%In \cite{key-3}, the countable concatenation property is broadly used as an effective tool. However, we introduce the weak countable concatenation property that fit better in many cases. 
%~\\

%It is obvious that every $L^0$-module is closed under countable concatenation. However, not every $L^0$-module has the countable concatenation property. To illustrate this, we shall see the following example.
%
%\begin{example}
%\label{exa: exampleNoCCP}
%Given a probabilistic space $(\Omega,\mathcal{F},\PP)$ and an infinite partition $\left\{ A_{n}\right\}_{n\in\mathbb{N}}$
%of $\Omega$ with $A_{n}\in\mathcal{F}$ and $P(A_n)>0$ (for example, $\Omega=(0,1)$, $\mathcal{F}=\mathcal{B}(\Omega)$ the $\sigma$-algebra of Borel, $A_{n}=[\frac{1}{2^n},\frac{1}{2^{n-1}})$ with $n\in\mathbb{N}$ and $P$ the Lebesgue measure). 
%~\\
%
%Let 
%\[
%E:=\left\{ \sum_{i\in F}\alpha_{i}1_{A_{i}};\: F\subset\mathbb{N}\textnormal{ is finite and }\alpha_{i}\in L^0(\Omega,\mathcal{F},\PP)\right\}.
%\]
%~\\
%
%Then $E$ is a $L^{0}$-module which does not have the countable concatenation property.
%~\\
%
%Define
%\[
%K:=\left\{X\in E ; \: |X|\leq 1 \right\}.
%\]
%
%Then $K$ is a proper subset that is closed under countable concatenations, but does not have the countable concatenation property. 
%~\\
%
%On the other hand, $E$ is not closed under countable concatenations seen as a subset of $L^0$. 
 %
%\end{example} 

Since $L^0$ is not a totally ordered set, we need to take advantage of the notion of countable concatenation. We have the following result:

\begin{prop}
\label{prop: cccdownwardsDirected}
Let $C\subset L^0$ be bounded below (resp. above) and stable under countable concatenations,  then for each $\varepsilon\in L^0_{++}$ there exists $Y_\varepsilon \in C$ such that 
\[
\essinf C \leq Y_\varepsilon < \essinf C + \varepsilon
\]
 (resp., $\esssup C \geq Y_\varepsilon > \esssup C - \varepsilon$) 

In particular, given a $L^0$-module $E$ and $K\subset E$, which is $L^0$-convex, $L^0$-absorbent and stable under countable concatenations, we have that, for $\varepsilon\in L^0_{++}$, there exists $Y_\varepsilon \in L^0_{++}$ with $X \in {Y_\varepsilon}K$ such that $p_K(X)\leq Y_\varepsilon < p_K(X)+\varepsilon$. 
\end{prop}  
\begin{proof}

Firstly, let us see that $C$ is downwards directed. Indeed, given $Y,Y'\in C$, define $A:=(Y<Y')$. Then, since $C$ is stable under countable concatenations, $1_A Y + 1_{A^c} Y'=Y\wedge Y' \in C$. 

Therefore,  for $\varepsilon \in L^0_{++}$, there exists a decreasing sequence $\left\{Y_k\right\}_{k}$ in $C$ converging  to $\essinf C$ almost surely. 

Let us consider the sequence of sets
\[
A_0:=\emptyset \textnormal{ and } A_k:=(Y_k<\essinf C +\varepsilon)-A_{k-1} \textnormal{ for }k>0. 
\]
Then $\left\{A_k\right\}_{k\geq 0}\in\Pi(\Omega,\mathcal{F}^+)$, and we can define $Y_\varepsilon:=\sum_{k\geq 0}{1_{A_k}}Y_k$. Given that $C$ is stable under countable concatenations, it follows that $Y_\varepsilon \in C$.
%\[
%Y_\varepsilon:=\sum_{k\geq 0}{1_{A_k}}Y_k.
%\]
%Thus $Y_\varepsilon \in C$, since $C$ is stable under countable concatenations.


For the second part, it suffices to see that if $K$ is stable under countable concatenations then $\left\{Y\in L^0_{++}; \: X\in {Y}K  \right\}$ is stable under countable concatenations as well. Indeed, given $\left\{A_k\right\}_{k}\in\Pi(\Omega,\mathcal{F})$ and $\left\{Y_k\right\}_{k} \subset L^0_{++}$ such that $X\in {Y_k}K$ for each $k\in\N$, let us take $Y:={\sum}_{n\in\mathbb{N}}Y_{n}1_{A_{n}}\in L_{++}^{0}$. Then we have that $X/Y\in cc\left(\{A_{k}\}_k,\left\{X/Y_{k}\right\}_k\right)$ and $X/Y_{k}\in K$.
%\[
%X/Y\in cc\left(\{A_{k}\}_k,\left\{X/Y_{k}\right\}_k\right)
%\]

Since $K$ is stable under countable concatenations, we conclude that  $X/Y\in K$, and the proof is complete.  
\end{proof}

Now, we have the following proposition

\begin{prop}
\label{closureGauge}
Let $E\left[\tau\right]$ a topological $L^0$-module and 
$C\subset E$ a $L^{0}$-convex and $L^{0}$-absorbent subset.  Then the following are equivalent 
\begin{enumerate}
	\item $p_C:E\rightarrow L^0$ is continuous. 
	\item $0\in \interior{C}$
\end{enumerate}

In this case, if in addition $C$ has the relative countable concatenation property
\[
\overline{C}=\left\{X \in E ; \: p_C(X)\leq 1  \right\}.
\]
\end{prop}

\begin{proof}

The proof is exactly the same as the real case.


For the equality.

``$\subseteq$'': It is obtained from the continuity.


``$\supseteq$'': Let $X\in E$ be satisfying $p_C(X)\leq 1$. By proposition \ref{prop: cccdownwardsDirected}, we have that for every $\varepsilon \in L^0_{++}$ there exists $Y_\varepsilon \in L^0_{++}$ such that
\[
1 \leq Y_\varepsilon < 1+\varepsilon,
\]
with $X \in Y_\varepsilon C$.

Then, $\left\{X/Y_\varepsilon \right\}_{\varepsilon \in L^0_{++}}$ is a net in $C$ converging to $X$. Thus $X\in \overline{C}$. And the proof is complete.
\end{proof}

Let us see an example showing that for the equality proved in the last proposition it is necessary to take $C$ with the relative countable concatenation property. 
\begin{example}
Given  $\Omega=(0,1)$, $\mathcal{E}=\mathcal{B}(\Omega)$ the $\sigma$-algebra of Borel, $A_{n}=[\frac{1}{2^n},\frac{1}{2^{n-1}})$ with $n\in\mathbb{N}$, $\PP$ the Lebesgue measure and $E:=L^0(\mathcal{E})$ endowed with $|\cdot|$. 


We define the set 

\[
U:=\left\{ Y\in L^{0};\:\exists\, I\subset\mathbb{N}\textnormal{ finite, }\left|Y1_{A_{i}}\right|\leq 1\:\forall\, i\in\mathbb{N}-I\right\}.
\]


Then, it is easily shown that $U$ is
$L^{0}$-convex, $L^{0}$-absorbent and $\overline{U}=U$.


Nevertheless, it can be proved that $p_{U}\left(X\right)=0$ for all $X\in{L^0}$, and therefore
\[
\left\{ X\in E;\: p_{U}(X)\leq 1\right\} = E.
\]

\end{example}


\section{A random version of the Mazur's lemma}

Finally, let us turn to prove a version for $L^0$-modules of the classical Mazur's lemma.

We need a new notion:

\begin{defn}
Let $E$ be a $L^0$-module, we say that the sum of $E$ preserves the relative countable concatenation property, if for every $L,M$ subsets of $E$ with the relative countable concatenation property it holds that the sum of both $L+M$ has the relative countable concatenation property. 
\end{defn} 

%\begin{lem}
%
%\end{lem}

\begin{thm}
\label{thm: Mazur}
[Randomized version of the Mazur lemma]
Let $(E,\left\Vert \cdot\right\Vert)$ be a $L^0$-normed module whose sum preserves the relative countable concatenation property, and let $\{X_\gamma\}_{\gamma\in \Gamma}$ be a net in $E$, which converges weakly to $X\in E$. Then, for any $\varepsilon\in L^0_{++}$, there exists
\[
Z_{\varepsilon} \in \overline{co}^{cc}_{L^0}\{X_\gamma ; \:\gamma\in \Gamma \} \textnormal{ such that }\left\Vert X - Z_\varepsilon \right\Vert \leq \varepsilon.
\] 

%\[
%\left\Vert X - Z_\varepsilon \right\Vert \leq \varepsilon
%\]
\end{thm}

%\begin{rem} 
%In the earliest version of this paper, was accidentally omitted the hypothesis on $E$ of having sum which preserves the relative concatenation property. The author is grateful to Tiexin Guo for pointing out to him the possible lack of this condition. Also, some discussion with Tiexin Guo motivated Example \ref{exam: counterExSPrccp} and Proposition \ref{prop: counterExSPrccp}.



%Before proving this statement, we would like to highlight that we do not need to include the countable concatenation on $E$. We argue by way of contradiction, and the argument rely on the fact that the countable concatenation hull —which was defined without assuming the c. c. property on $E$— has the relative countable concatenation property. That is sufficient to reach a contradiction, by using the gauge function and the extension Hahn-Banach Theorem for $L^0$-modules, result that does not require the c. c. property on $E$ either. 
%\end{rem}

\begin{proof}
Define $M_1 := \overline{co}^{cc}_{L^0} \{X_\gamma ; \: \gamma\in \Gamma \}.$ We may assume that $0\in M_1$, by replacing $X$ by $X-X_{\gamma_0}$ and $X_\gamma$ by $X_\gamma-X_{\gamma_0}$ for some $\gamma_0\in \Gamma$ fixed and all $\gamma\in\Gamma$.

By way of contradiction, suppose that for every $Z \in M_1$ there exists $A_Z \in \mathcal{F}^+$  such that $\left\Vert X - Z \right\Vert > \varepsilon$ on $A_Z$.
%\begin{equation}
%\label{ineqI}
%\begin{array}{cc}
%\left\Vert X - Z \right\Vert > \varepsilon & \textnormal{ on } A_Z 
%\end{array}
%\end{equation}

Denote $B_\frac{\varepsilon}{2}:=\{ X\in E; \Vert X \Vert\leq \frac{\varepsilon}{2}\}$, and define $M:=\underset{Z\in M_1}\bigcup{Z + B_\frac{\varepsilon}{2}}.$

Then $M$ is a $L^0$-convex $L^0$-absorbent neighbourhood of $0\in E$, which has the relative countable concatenation property (this is because $M_1$ and $B_\frac{\varepsilon}{2}$ has the relative c. c. property and the sum of $E$ preserves the relative c. c. property). Besides, for every $Z \in M$ there exists $C_Z \in \mathcal{F}^+$ with $\Vert X-Z\Vert\geq\frac{\varepsilon}{2}$ on $C_Z$. So that $X \notin \overline{M}$.
%\begin{equation}
%\label{ineqII}
%\begin{array}{cc}
%\left\Vert X - Z \right\Vert > \frac{\varepsilon}{2} & \textnormal{ on } C_Z. 
%\end{array}
%\end{equation}

%So that $X \notin \overline{M}$.

Thus, by Proposition \ref{closureGauge} we have that there exists $C\in \mathcal{F}^+$ such that
\begin{equation}
\label{ineqII}
\begin{array}{cc}
p_M (X) > 1 & \textnormal{ on } C,
\end{array}
\end{equation}

where $p_M$ is the guage function of $M$.

Further, given $Y,Y' \in L^0$ with $1_C Y X =1_C Y' X$, it holds that $Y=Y'$ on $C$. 

Indeed, define $A=(Y-Y'\geq 0)$
\[
1_C |Y-Y'| p_M (X)\leq p_M (1_C |Y-Y'|X)=p_M ((1_{A}-1_{A^c})1_C (Y-Y')X)=p_M(0)=0.
\]
In view of \ref{ineqII}, we conclude that $Y=Y'$ on $C$.  

Then, we can define the following $L^0$-linear application 
\[
\begin{array}{cc}
\mu_0 : & span_{L^0} \{X\}\longrightarrow L^0\\
& \mu_0 (Y X):=Y 1_{C} p_M(X).
\end{array}
\]


In addition, we have that
\[
\begin{array}{cc}
\mu_0 (Z) \leq p_M (Z) & \textnormal{ for all } Z\in span_{L^0}\{X\}.
\end{array}
\] 

Thus, by the Hahn-Banach extension theorem for $L^0$-modules (\cite{key-3}, theorem 2.14), there exists a $L^0$-linear extension $\mu$ of $\mu_0$ defined on $E$ such that 
\[
\begin{array}{cc}
\mu (Z) \leq p_M (Z) & \textnormal{ for all } Z\in E.
\end{array}
\]
 
Since $M$ is a neighborhood of $0\in E$, by Proposition \ref{closureGauge}, the gauge function $p_M$ is continuous on $E$. Hence $\mu$ is a continuous
$L^0$-linear function defined on $E$.


Furthermore, we have that
\[
\underset{Z \in M_1} \esssup \mu (Z) \leq \underset{Z \in M} \esssup \mu (Z) \leq
\]
\[
\begin{array}{cc}
\leq \underset{Z \in M} \esssup p_M (Z) \leq 1 < p_M (X)=\mu(X)  & \textnormal{ on }C. 
\end{array}
\]
Therefore, $X$ cannot be a weak accumulation point of $M_1$ contrary to the hypothesis of $X_\gamma$ converging weakly to $X$.
\end{proof}



We have the following corollaries: 


\begin{cor}
\label{prop: closedw}
Let $(E,\left\Vert \cdot\right\Vert)$ be a $L^0$-normed module whose sum preserves the relative countable concatenation property, and let $K\subset E$  be $L^0$-convex and with the relative countable concatenation property, we have that the closure in norm coincides with the closure in the weak  topology, i.e. $\overline{K}^{\Vert \cdot \Vert} = \overline{K}^{\sigma(E,E^*)}$.
\end{cor}

Then, from now on, for any subset $K$ which is $L^0$-convex and with the relative countable concatenation property, we will denote the topological closure by $\overline{K}$ without specifying whether the topology is either weak or strong.

Let us recall some notions:

\begin{defn}
Let $E[\tau]$ be a topological $L^0$-module. A function $f:E\rightarrow\bar{L}^{0}$ is called proper if $f(E)\cap L^0 \neq \emptyset$ and $f>-\infty$. It is said to be $L^0$-convex if $f (Y X_1 + (1 − Y)X_2) \leq Y f (X_1) + (1 − Y)f (X_2)$ for all $X_1,X_2 \in E$ and
$Y \in L^0$ with $0 \leq Y \leq 1$. It said to have the local property if $1_A f(X)=1_A f(1_A X)$ for $A\in\mathcal{F}^+$ and $X\in E$. Finally, $f$ is called lower semicontinuous if the level set $V\left(Y_{0}\right)=\left\{ X\in E;\: f\left(X\right)\leq Y_{0}\right\}$ is closed for all $Y_0\in L^0$.
\end{defn}

\begin{cor}
\label{cor: lowerSemCont}
Let $(E,\left\Vert \cdot\right\Vert)$ be a $L^0$-normed module whose sum preserves the relative countable concatenation property, and let $f:E\rightarrow \bar{L}^0$  be a proper $L^0$-convex function. If $f$ is continuous, then $f$ is lower semicontinuous with the weak topology. 
\end{cor}




\begin{proof}
It is a known fact that, if $f$ is $L^0$-convex, then it has the local property (see \cite[Theorem 3.2]{key-8}).

Being $f$ $L^0$-convex and with the local property, we have that $V(Y)$ is $L^0$-convex and has the relative countable concatenation property.

Since $f$ is continuous, we have that $V(Y)$ is closed, and due to  Corollary \ref{prop: closedw}, it is weakly closed as well.

\end{proof}




Finally, we will provide an example showing that in the version of Mazur's lemma proved, rather than just take $X_{\varepsilon}$ into the $L^0$-convex hull, we must take it into the countable concatenation closure of the $L^0$-convex hull. Namely, we shall give an example of a net weakly convergent to some limit, which is not a cluster point of the $L^0$-convex hull of that net.   

\begin{example}
Given  $\Omega=(0,1)$, $\mathcal{E}=\mathcal{B}(\Omega)$ the $\sigma$-algebra of Borel, $A_{n}=[\frac{1}{2^n},\frac{1}{2^{n-1}})$ with $n\in\mathbb{N}$ and $\PP$ the Lebesgue measure. 
~\\

We define
\[
\mathcal{F}:=\sigma(\left\{A_n ; \: n\in \N \right\})
\]
the $\sigma$-algebra generated by $\left\{A_n ; \: n\in \N \right\}$.
~\\

Then, we take the $L^0(\mathcal{F})$-module
\[
L_{\mathcal{F}}^{2}\left(\mathcal{E}\right):=L^{0}\left(\mathcal{F}\right)L^{2}\left(\mathcal{E}\right)
\] 
and the $L^0(\mathcal{F})$-seminorm
\[
\left\Vert X \mid \mathcal{F} \right\Vert_{2}:=\E\left[\left|X\right|^{2}|\mathcal{F}\right]^{1/2} 
\]
as we can see defined in \cite{key-3}.
~\\

Then the following holds
\begin{equation}
\label{eq1}
L^{0}\left(\mathcal{F}\right)=\left\{\sum_{n\in \N} \alpha_n 1_{A_n} ; \: \alpha_n \in \R \right\}
\end{equation}
\begin{equation}
\label{eq2}
L_{\mathcal{F}}^{2}\left(\mathcal{E}\right)=\left\{\sum_{n\in \N} X_n 1_{A_n} ; \: X_n \in L^{2}\left(\mathcal{E}\right) \right\}
\end{equation}
\begin{equation}
\label{eq3}
\begin{array}{cc}
\left\Vert X \mid \mathcal{F} \right\Vert_{2}^2:=\sum_{n\in \N} \frac{\Vert X 1_{A_n}\Vert_2^2}{1/2^n}1_{A_n}
 & \textnormal{ for } X\in L_{\mathcal{F}}^{2}\left(\mathcal{E}\right).
\end{array}
\end{equation}

Now, we will define a net in $L_{\mathcal{F}}^{2}\left(\mathcal{E}\right)$ indexed with the set $\N ^ \N$. Given $\{ n_k\}_{k\in\N}$ we define
\[
\begin{array}{cc}
X_{\{ n_k\}_{k\in\N}} (t):=\sum_{k\in\N} 1_{A_k} \sgn [\sin 2\pi (2^{k+n_k} t - 1)] & \textnormal{ for } t\in (0,1).
\end{array}
\]


We shall show that this net converges weakly to $0$ and that $0$ is not a cluster point of $co_{L^0}\left\{X_{\{n_k\}} ; \: \{n_k\}\in \N^\N \right\}$.
~\\

Indeed, by \cite{key-12} or \cite{key-10}, we know that for each $X^* \in E^*$, there exists $Y\in L_{\mathcal{F}}^{2}\left(\mathcal{E}\right)$ such that 
\[
\left\langle X, X^* \right\rangle=\E\left[X{Y}|\mathcal{F}\right]
\]


But, for every $Y\in L_{\mathcal{F}}^{2}\left(\mathcal{E}\right)$ 
\[
\left| \E\left[X_{\{n_k\}}{Y}|\mathcal{F}\right] \right|=\underset{k\in\N}\sum{\frac{\E[1_{A_k}{Y}X_{n_k}]}{\PP(A_k)}1_{A_k}}=
\]

\[
=\underset{k\in\N}\sum{\frac{\left| \int_{1/2^k}^{1/2^{k-1}}{{Y}\sgn[\sin {2\pi (2^{k+{n_k}} s - 1  )}]} d s \right|}{1/{2^k}}1_{[\frac{1}{2^k},\frac{1}{2^{k-1}})}},
\]

and it can be proved that the later converges to $0$ on $A_k$ for $k=1,2,...$. Hence, since $Y$ is arbitrary we conclude that $X_{\{n_k\}}$ converges weakly to $0$.
~\\

On the other hand, let us see that $0$ is not a cluster point of $co_{L^0}\left\{X_{\{n_k\}} ; \: \{n_k\}\in \N^\N \right\}$.
~\\

Indeed, given $Y\in co_{L^0}\left\{X_{\{n_k\}} ; \: \{n_k\}\in \N^\N \right\}$. We have that $Y$ will be as follows 
\[
%\begin{array}{cc}
Y=\sum_{k\in \N} 1_{A_k} \sum_{i=1}^N{\alpha_k^i \sgn[\sin {2\pi (2^{k+{n_k}} s - 1  )}]} %& \textnormal{ with } \alpha_k\in \R,0\leq  \leq N k=1,2,...N.
%\end{array}
\]

with $N\in \N$, $\alpha_k^i\in\R$ and with $\sum_{i=1}^N{\alpha_k^i}=1$ for all $k\in\N$.

In addition, we have that it can be proved that
\[
\left\Vert 1_{A_k} \sum_{i=1}^N{\alpha_k^i \sgn[\sin {2\pi (2^{k+{n_k}} s - 1  )}]} \right\Vert_2^2 \geq 
\]
\[
\geq \PP\left(A_k \cap \bigcap_{j=1}^N \left({\sgn[\sin {2\pi (2^{k+{n_k}} s - 1  )}]=1}\right) \right) \geq \frac{1}{2^{N+K-1}}.
\]

Therefore, by using \ref{eq3}
\[
\left\Vert Y \mid \mathcal{F} \right\Vert_{2}^2 = \sum_{k\in \N}{\frac{\left\Vert 1_{A_k} \sum_{i=1}^N{\alpha_k^i \sgn[\sin {2\pi (2^{k+{n_k}} s - 1  )}]} \right\Vert_2^2}{1/2^k}1_{A_k}}\geq
\]
\[
=\sum_{k\in \N} \frac{1/2^{N+k-1}}{1/2^k} 1_{A_k}= \sum_{k\in \N} \frac{1}{2^{N-1}} 1_{A_k}.
\]

But, taking $\varepsilon:=\sum_{k\in \N} \frac{1}{2^{k}} 1_{A_k} \in L^0_{++}(\mathcal{F})$ it is clear that for each $Y\in co_{L^0}\left\{X_{\{n_k\}} ; \: \{n_k\}\in \N^\N \right\}$ there exists $A\in\mathcal{F}$ with $\PP(A)>0$ such that
\[
\begin{array}{cc}
\left\Vert Y \mid \mathcal{F} \right\Vert_{2}^2 > \varepsilon & \textnormal{ on } A.
\end{array}
\]
Hence, $0$ cannot be a cluster point of $co_{L^0}\left\{X_{\{n_k\}} ; \: \{n_k\}\in \N^\N \right\}$ as could be expected considering the classical Mazur's lemma. 
\end{example}



%\begin{thm}
%\label{thm: caracterizacion}
%Let $E\left[\tau\right]$ be a topological $L^{0}$-module. Then $\tau$ is induced by a family of $L^0$-seminorms if, and only if, there is a neighborhood base of $0\in{E}$ for which each $U\in\mathcal{U}$ is 
%\begin{enumerate}
%\item $L^{0}$-convex,
%\item $L^{0}$-absorbent,
%\item $L^{0}$-balanced and
%\item closed under countable concatenations, i.e, $U=\overline{U}^{cc}$.
%\end{enumerate}
%\end{thm}
%
%\begin{proof}
%Suppose that $\tau$ is induced by a family of $L^0$-seminorms. 
%If $Q\subset\mathcal{P}$ is finite and $\varepsilon\in L_{++}^{0}$, by inspection follows that $U_{Q,\varepsilon}$ is $L^{0}$-convex, $L^{0}$-absorbent and $L^{0}$-balanced.
%Besides, $U_{Q,\varepsilon}$ is closed under countable concatenations.
%Indeed, if $X=\sum_{n}1_{A_{n}}X_{n}$ with $X_{n}\in U_{Q,\varepsilon}$
%for all $n\in\mathbb{N}$ and $\left\{ A_{n}\right\} _{n\in\mathbb{N}}$
%is a partition of $\Omega$ with $A_n\in\mathcal{F}$ it holds for
 %$\left\Vert \cdot\right\Vert \in Q$ that
%\[
%\left\Vert X\right\Vert =\left(\sum_{n}1_{A_{n}}\right)\left\Vert X\right\Vert =\sum_{n}1_{A_{n}}\left\Vert X\right\Vert =
%\]
%
%
%\[
%=\sum_{n}\left\Vert 1_{A_{n}}X\right\Vert =\sum_{n}\left\Vert 1_{A_{n}}X_{n}\right\Vert =\sum_{n}1_{A_{n}}\left\Vert X_{n}\right\Vert \leq\varepsilon.
%\]
%
%
%Reciprocally, let $\mathcal{U}$ be a neighborhood base of $0\in{E}$ for which each $U\in\mathcal{U}$ is 
%$L^{0}$-convex, $L^{0}$-absorbent, $L^{0}$-balanced and closed under countable concatenations.
%Let us consider the family of $L^0$-seminorms $\left\{ p_{U}\right\} _{U\in\mathcal{U}}$ and let us show that it induces the topology $\tau$.
 %Given $U\in\mathcal{U}$ is clear that $U\subset U_{p_{U},1}$. Therefore, for $\varepsilon\in L_{++}^{0}$ there exists $U\text{'\ensuremath{\in\mathcal{U}}}$ such that $\frac{1}{\varepsilon}U'\subset U\subset U_{p_{U},1}$.
%Thus, $U'\subset U_{p_{U},\varepsilon}$. 
%On the other hand, for $U\in\mathcal{U}$, it holds that $U_{p_{U},\frac{1}{2}}\subset\left\{ X\in E;\: p_{U}(X)<1\right\} \subset U$.
%\end{proof}
%
%Taking advantage of the ideas of the last theorem, we provide an example of a locally $L^0$-convex topology not induced by any family of $L^0$-seminorms.
%
%\begin{example}
%\label{contraejemplo}
%Given a probabilistic space $(\Omega,\mathcal{F},\PP)$ and an infinite partition $\left\{ A_{n}\right\}_{n\in\mathbb{N}}$
%of $\Omega$ with $A_{n}\in\mathcal{F}$ and $P(A_n)>0$ (for example, $\Omega=(0,1)$, $\mathcal{F}=\mathcal{B}(\Omega)$ the $\sigma$-algebra of Borel, $A_{n}=[\frac{1}{2^n},\frac{1}{2^{n-1}})$ with $n\in\mathbb{N}$ and $P$ the Lebesgue measure). 
%~\\
%
%Let $E=L^0$.
%~\\
%
%For $\varepsilon\in L_{++}^{0}$, we define the set 
%
%\[
%U_{\varepsilon}:=\left\{ Y\in L^{0};\:\exists\, I\subset\mathbb{N}\textnormal{ finite, }\left|Y1_{A_{i}}\right|\leq\varepsilon\:\forall\, i\in\mathbb{N}-I\right\}.
%\]
%
%
%Then, it is easily shown that $U_{\varepsilon}$ is
%$L^{0}$-convex, $L^{0}$-absorbent and $L^{0}$-balanced, and
%$\mathcal{U}:=\left\{ U_{\varepsilon};\:\varepsilon\in L_{++}^{0}\right\} $
%is a neighborhood base of $0\in{E}$ which generates a topology for which $L^{0}$ is a locally $L^{0}$-convex module.
%
%Furthermore, it holds that $U_{\varepsilon}$ is not closed under countable concatenations.
%
%
%Indeed, it is verified that $\varepsilon+1\notin U_{\varepsilon}$, but
%$\varepsilon+1=\sum_{n}\left(\varepsilon+1\right)1_{A_{n}}$ with $\left(\varepsilon+1\right)1_{A_{n}}\in U_{\varepsilon}$.
%
%Easily, it can be shown that any neighborhood base of $0\in{E}$ generating the same topology verified that its elements are not closed under countable concatenations on $L^{0}$.
%
%Therefore, due to theorem \ref{thm: caracterizacion}, $L^{0}$ endowed with the topology generated by $\mathcal{U}$, is a locally $L^0$-convex topology, which is not induced by any family of $L^0$-seminorms.
%
%Besides, it has to be met that $\left\{ X\in L^{0};\: p_{U_{\varepsilon}}\left(X\right)<1\right\} \nsubseteq U_{\varepsilon}$
%for some $\varepsilon\in L_{++}^{0}$.  Otherwise, the family of $L^0$-seminorms
 %$\left\{ P_{U_{\varepsilon}}\right\}_{\varepsilon\in L_{++}^{0}}$ would induce the topology.
%
%
%In fact, we claim that $p_{U}\left(X\right)=0$ for all $X\in{L^0}$ and $U\in\mathcal{U}$.
%It suffices to show that $p_{U_1}\left(1\right)=0$, since $p_{U_1}$ is a $L^0$-seminorm. By way of contradiction, assume $p_{U_1}\left(1\right)>0$. Then, there exists $m\in\N$ 
%such that $P\left[\left(p_{U_1}\left(1\right)>0\right)\cap{A_m}\right]>0$.
%Define $A:=\left(p_{U_1}\left(1\right)>0\right)\cap{A_m}$, $\nu:=\frac{p_{U_1\left(1\right)}+1_{A^c}}{2}$, $Y:=1_{A^c}+\nu{1_A}$ and $X:=1_{A^c}+\frac{1}{\nu}{1_A}$. 
%Thus, we have $1=Y{X}\in{YU_1}$ and $P\left(p_{U_1}\left(1\right)>Y\right)>0$. We have a contradiction. 
%
%\end{example}
%
%Theorem \ref{thm: caracterizacion} gives rise to give a new more restrictive definition of locally $L^0$-convex module.
 %In the following, a locally $L^0$-convex module will be as definition below says.
%
%
%\begin{defn}
%A locally $L^0$-convex module is a $L^{0}$-module such that there is a neighborhood base of $0\in{E}$ for which each $U\in\mathcal{U}$ is 
%\begin{enumerate}
%\item $L^{0}$-convex,
%\item $L^{0}$-absorbent,
%\item $L^{0}$-balanced and
%\item closed under countable concatenations.
%\end{enumerate}
%\end{defn}
%
%Thus, under this new definition a $L^{0}$-module $E\left[\tau\right]$ is a locally $L^0$-convex module if, and only if, $\tau$ is induced by a family of $L^0$-seminorms.
%
%
%\begin{prop}
%\label{prop: caracterizacion}
%A $L^0$-module $E\left[\tau\right]$ is locally $L^0$-convex if, and only if, $\tau$ is induced by a family of $L^0$-seminorms.
%\end{prop}



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\bibitem{key-5}Frittelli, M., and Maggis, M. (2014). \emph{Complete duality for quasiconvex dynamic risk measures on modules of the Lp-type.} Statistics \& Risk Modeling, 31(1), 103-128.
\bibitem{key-12} Guo, T. (2013). \emph{On Some Basic Theorems of Continuous Module Homomorphisms between Random Normed Modules}. Journal of Function Spaces and Applications.
\bibitem{key-6}Guo, T. (2011). \emph{Recent progress in random metric theory and its applications to conditional risk measures.} Science China Mathematics, 54(4), 633-660.
\bibitem{key-7}Guo, T. (2010). \emph{Relations between some basic results derived from two kinds of topologies for a random locally convex module.} Journal of Functional Analysis, 258(9), 3024-3047.
\bibitem{key-8}Guo, T., Zhao, S., and Zeng, X. (2012). \emph{On random convex analysis-the analytic foundation of the module approach to conditional risk measures.} arXiv preprint arXiv:1210.1848.
\bibitem{key-9}Guo, T., Zhao, S., and Zeng, X. \emph{The relations among the three kinds of conditional risk measures.} Science China Mathematics, 1-12.
\bibitem{key-10}Kupper, M., and Vogelpoth, N. (2009). \emph{Complete $L^0$-normed modules and automatic continuity of monotone $L^0$-convex functions.} Preprint.
\bibitem{key-11} Yosida, K. (1980) \emph{Functional Analysis}. Spring-Verlag, New York/Berlin (1971).
\bibitem{key-13} Zapata, J. M. (2017). \emph{On characterization of locally $L^0$-convex topologies induced by a family of $L^0$-seminorms.} Journal of Convex Analysis 24.

%
%
%\bibitem{key-5} N. Dunford and J. T. Schwartz, \emph{Linear Operators I}, Interscience, New York, NY, USA, 1957.
%\bibitem{key-6} R. Haydon, M. Levy, and Y. Raynaud, \emph{Randomly Normed Spaces}, Hermann, Paris, France, 1991.
%\bibitem{key-7} B. Schweizer, A. Sklar, \emph{Probabilistic Metric Spaces}, Elsevier/North-Holland, New York, 1983; reissued by Dover Publications, Mineola, New York, 2005.
%\bibitem{key-7} T. Guo, \emph{A comprehensive connection between the basic results and properties derived from two kinds of topologies for a random locally convex module.} arXiv preprint arXiv:0908.1843 (2009).

 
\end{thebibliography}
\end{document}

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{key-5Frittelli, M., and Maggis, M. (2014). Complete duality for quasiconvex dynamic risk measures on modules of the Lp-type. Statistics \& Risk Modeling, 31(1), 103-128.
1409.7932.extracted_bib
{key-12 Guo, T. (2013). On Some Basic Theorems of Continuous Module Homomorphisms between Random Normed Modules. Journal of Function Spaces and Applications.
1409.7932.extracted_bib
{key-6Guo, T. (2011). Recent progress in random metric theory and its applications to conditional risk measures. Science China Mathematics, 54(4), 633-660.
1409.7932.extracted_bib
{key-7Guo, T. (2010). Relations between some basic results derived from two kinds of topologies for a random locally convex module. Journal of Functional Analysis, 258(9), 3024-3047.
1409.7932.extracted_bib
{key-8Guo, T., Zhao, S., and Zeng, X. (2012). On random convex analysis-the analytic foundation of the module approach to conditional risk measures. arXiv preprint arXiv:1210.1848.
1409.7932.extracted_bib
{key-9Guo, T., Zhao, S., and Zeng, X. The relations among the three kinds of conditional risk measures. Science China Mathematics, 1-12.
1409.7932.extracted_bib
{key-10Kupper, M., and Vogelpoth, N. (2009). Complete $L^0$-normed modules and automatic continuity of monotone $L^0$-convex functions. Preprint.
1409.7932.extracted_bib
{key-11 Yosida, K. (1980) Functional Analysis. Spring-Verlag, New York/Berlin (1971).
1409.7932.extracted_bib
{key-13 Zapata, J. M. (2017). On characterization of locally $L^0$-convex topologies induced by a family of $L^0$-seminorms. Journal of Convex Analysis 24. % % %
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{key-5 N. Dunford and J. T. Schwartz, Linear Operators I, Interscience, New York, NY, USA, 1957. %
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{key-6 R. Haydon, M. Levy, and Y. Raynaud, Randomly Normed Spaces, Hermann, Paris, France, 1991. %
1409.7932.extracted_bib
{key-7 B. Schweizer, A. Sklar, Probabilistic Metric Spaces, Elsevier/North-Holland, New York, 1983; reissued by Dover Publications, Mineola, New York, 2005. %
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{key-7 T. Guo, A comprehensive connection between the basic results and properties derived from two kinds of topologies for a random locally convex module. arXiv preprint arXiv:0908.1843 (2009).