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\fancyhead[CE]{S. Mohammadzadeh} 
\fancyhead[CO]{Specific Complete Measure in the Structure of a Utility Function}



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{\noindent Journal of Mathematical Extension \\
Vol. XX, No. XX, (2014), pp-pp (Will be inserted by layout editor)}\\
ISSN: 1735-8299\\
URL: http://www.ijmex.com\\
\vspace*{9mm}

\begin{center}

{\Large \bf 
Specific Complete Measure in the Structure of a Utility Function\\}


\let\thefootnote\relax\footnote{\scriptsize Received: XXXX; Accepted: XXXX (Will be inserted by editor)}

{\bf Somayeh Mohammadzadeh$^*$}\vspace*{-2mm}\\
\vspace{2mm} {\small  University of Bojnord} \vspace{2mm}

\end{center}

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{\footnotesize
\begin{quotation}
{\noindent \bf Abstract.} An outer measure is constructed on a pseudo-ordered set $(   X , \succsim )$ and then it will be shown that it is in fact a measure  defined on the whole  power set of $X$ . Applying this, a measurable  utility function $\theta$ is defined which represents the  relation $\succsim$ on $X$. Also, we discuss the continuity  and upper semi-continuity of $\theta$ in certain points of $X$.
 Finally, the results are used to improve some of the theorems in economics.
\end{quotation}
\begin{quotation}
\noindent{\bf AMS Subject Classification:} Primary 28C15, Secondary 28A12

\noindent{\bf Keywords and Phrases:} measure, utility function, continuous, upper semi-continuous
\end{quotation}}

\section{Introduction}
\label{intro} % It is advised to give each section and subsection a unique label.
Suppose that  $( X,\succsim ) $ is a pseudo-ordered space; that is, $\succsim$ is a binary relation on $X$ with two following properties:

$(i)$ Completeness: For all $x,y\in X$, either $x \succsim y$ or $y \succsim x$;

$(ii)$ Transitivity:  For all $x,y,z\in X$, if $x \succsim y$ and $y \succsim z$, then $x \succsim z$.\\
The  relation $x\succ y$ means   $x \succsim y$ , but not  $y \succsim x$. Also $x \thicksim y$ means  $x \succsim y$ and  $y \succsim x$.\\
Let $x_{0}$ be any point in $X$. Relative to any such point, we can define
the following subsets of X:
\bea W(  x_{0})   &=& \left\lbrace  x\in X; x_{0}\succ x \right\rbrace ,\\
B(  x_{0})   &=& \left\lbrace  x\in X; x \succ x_{0} \right\rbrace , \\ 
I\left(  x_{0}\right)   &=& \left\lbrace  x\in X; x \thicksim x_{0} \right\rbrace .\eea
The topology on $X$ generated by the collections \[\{ W(  y ) ; y\in X\} \  \textrm{and} \  \{ B(  y ) ; y\in X\}\] is called the order topology.
% By this topology the closure of $W(  x_{0})$ and $B(  x_{0})$ are
%$$\overline{W}\left(  x_{0}\right)   = \left\lbrace  x\in X; x_{0} \succsim x \right\rbrace  \  \textrm{and}$$
%$$\overline{B}\left(  x_{0}\right)   = \left\lbrace  x\in X; x  \succsim x_{0} \right\rbrace $$ respectively.
For each  $x,y\in X$ the  open interval $\left( x , y\right)$  is defined by
\[ \left( x , y\right) = \left\lbrace z\in X; x\prec z \prec y\right\rbrace .\]
The other intervals $\left[ x , y\right], \left( x , y\right]$ and $\left[ x , y\right)$  are defined similarly. \\
In the following, we will assume that  $X$ has a minimum element and it is unbounded from above. \\
A real-valued function $u: X\longrightarrow \mathbb{R} $ is called a utility  function representing the
relation $\succsim$, if it is an strictly increasing function with respect to the
relation $\succsim$; that is, for all $x,y\in X$,
\bea x\thicksim y & \Rightarrow & u( x )  = u( y ),\\
x\succ y & \Rightarrow & u( x )  > u( y ).\eea
A function  $u: X\longrightarrow \R $ is said to be upper semi-continuous if \[\{x\in X;  u( x ) < r \}\] is open for each $r \in \R$.\\
The relation $\succsim$ on $X$ is called order separable if there is a countable set $D\subseteq X$ such that for all  $x,y\in X$,
\[ x\succ y \Rightarrow(  \exists \  d_{1}, d_{2}\in D; x\succsim d_{1} \succ d_{2} \succsim y).  \]
% Indeed $D$ is dense in $X$ with respect to the order topology.
 For every $D\subseteq X$, the topology  on $X$ generated by the family $\{ W(  y ) ; y\in D\}$ is called $D$-lower order topology. Also the topology generated by
 \[\{ W(  y ) ; y\in D\} \ \textrm{and} \  \{ B(  y ) ; y\in D\}\] is called $D$-order topology.

  In several discussions of economics, existence of a continuous ( or upper semi-continuous ) utility function
  on a pseudo-ordered space  $( X,\succsim ) $ has been studied. Among them,  \cite{j} and  \cite{ s} can be mentioned.
  Also in \cite{vw}, Voorneveld and Weibull constructed a utility function based on specific  outer measure; they proved the upper semi-continuity of this function in $D$-lower order topology of $X$. By following this method, in \cite{mk}, the combination of measure and utility theory have been used to represent an order relation.

  Continuing these works, in a manner similar to \cite{vw}, an outer measure $\nu$ is constructed. Then we show that it is in fact a measure on all power set of $X$. In the last section, as an application of this measure function in economic topics, we define a utility function based on $\nu$ and discuss about the continuity and upper semi-continuity of this function on some points of $X$.

\section{Creating a new measure}
\label{sec:2}
Suppose  $\succsim$ is an order separable relation on $X$. The set $D$ in the definition of order separability is countable, so we can give each $d\in D$ a positive weight, $\omega( d ) $, such that weights have a finite sum. Put  $\varepsilon_{t ( d ) } = \omega( d )$ in which $t: D \longrightarrow \N$ is an injection. Without loss of generality we assume that \[\sum_{d\in D} \omega( d ) =\sum_{d\in D} \varepsilon_{t ( d ) } = \sum _{i=1}^{\infty} \varepsilon_{i}=1;\]
  see  \cite{ vw}.
Let\[ \mathrm{F}= \{ W(  d ) ; d\in D\} \cup \{X, \emptyset \}\]
and define $\rho : \mathrm{F} \longrightarrow\left[ 0,1\right] $ with $\rho( \emptyset) = 0,  \rho( X) = 1$ and
\[\rho( W(  d )) = \sum_{d^{'}\in D; d^{'}\precsim d} \omega( d^{'} ) \ \ \ \forall d\in D \]
Applying ~\cite[Proposition 1.10]{f}, the function $\rho$ can be extended to an outer measure $\mu^{*}$ defined by
\bea \mu^{*}( A)&=&\inf \{ \sum_{i}\rho ( W_{i}); W_{i}\in  \F \ and \  A\subseteq\bigcup_{i\in \mathbb{N} } W_{i}\}\\&=&
\inf \{ \sum_{i} \sum_{{d^{'}\in D; d^{'}\precsim d_{i}}}  \omega( d^{'} ) ; W_{i}= W(  d_{i} )\in  \mathrm{F} \ and \  A\subseteq\bigcup_{i\in \N } W_{i}\}.\eea
Recall that a set $A\subseteq X $ is called  $\mu^{*}$- measurable if for each  $E\subseteq X $,
\[ \mu^{*}( E) =  \mu^{*}( E \cap A) +  \mu^{*}( E \cap A^{c}). \]
By the Caratheodory's theorem, ~\cite[Theorem 1.11]{f}, the collection $\mathcal{M}$ of  $\mu^{*}$-measurable sets is a $\sigma$-algebra and the restriction of
$\mu^{*}$ to $\mathcal{M}$, denoted by $\mu$, is a complete measure.The elements of  $\M$ are called measurable sets. \\
Now define  the function  $u: X\longrightarrow \R $ with
\[ u( x) = \mu^{*}( W( x) ) \ \ \ \forall x\in X.\]
The following theorem of \cite{ vw} is based on the definition of $u$ as an outer measure function.
\begin{theorem}~\cite[Theorem 3.1]{vw} Let $\succsim$ be order separable relation on $X$. The function $u$, defined as above, is a utility function represents
$\succsim$ and is upper semi-continuous in the $D$- lower order topology.
\end{theorem}
Maybe, extension of $\mu^{*}$ to a complete measure give some better results about the utility function $u$. But, in this case the structure of the  $\sigma$-algebra  $\M$   is not clear.
It seems that in some cases  $\mathcal{M}= \{X , \emptyset\}$.

Let $F$ be as above. Continuing in this argument, we define another outer measure $\nu$ in a similar way with different properties.
 Suppose that $\nu(\emptyset) = 0$ , $\nu(X)=1$ and for every proper subset $A$ of $X$ define
\[\nu( A):=\inf \{ \sum_{i} \sum_{{d^{'}\in D\cap A; d^{'}\precsim d_{i}}}  \omega( d^{'} ) ; W_{i}= W(  d_{i} )\in  \F \ \textrm{and} \  A\subseteq \bigcup_{i\in \N } W_{i}\}.\]
Without loss of generality we consider the set $D$ consists of the minimum of $X$ which will be denoted by $\tilde{d}$. In this case we assume that $\omega ( \tilde{d}) =0$ and hence
$\nu(W (\tilde{d}))=0$.
\begin{proposition}
The function $\nu$ defines an outer measure on $X$.
\end{proposition}
\begin{proof}
Suppose that $A\subseteq B \subseteq X$. Then  $A\subseteq\cup W_{i}$ for each sequence $\{W_{i}\}$ in $\F$ with $B\subseteq\cup W_{i}$. Thus
\bea \nu(A) & \leq & \sum_{i} \sum_{{d^{'}\in D\cap A; d^{'}\precsim d_{i}}}  \omega( d^{'} ) \\&\leq &
\sum_{i} \sum_{{d^{'}\in D\cap B; d^{'}\precsim d_{i}}}  \omega( d^{'} )\eea
 for every such sequence  $\{W_{i}\}$  and so $\nu(A) \leq \nu(B)$.\\
Now let $\{A_{t}\}$ be a sequence of subsets of $X$ and for each $t\in \N$ let
\[\nu( A_{t} ) =\inf \{ \sum_{i} \sum_{{d^{'}\in D\cap A_{t}; d^{'}\precsim d_{i}^{t}}}  \omega( d^{'} ) ; W_{i}^{t}= W(  d_{i}^{t} )\in  \F \ and \  A_{t}\subseteq\bigcup_{i\in \N } W_{i}^{t}\}.\]
Then for every $\varepsilon > 0$ there exist a sequence $\{W_{i}^{t}\}_{i\in \N}$ such that
\[ \sum_{i} \sum_{{d^{'}\in D\cap A_{t}; d^{'}\precsim d_{i}^{t}}}  \omega ( d^{'} )\leq \nu( A_{t} ) + \dfrac{\varepsilon}{2^{t}}.\]
Therefore
\bea \sum_{i} \sum_{{d^{'}\in D\cap(\cup_{t}A_{t}); d^{'}\precsim d_{i}^{t}}}  \omega ( d^{'} ) &\leq &  \sum_{i}  \sum_{t} \sum_{{d^{'}\in D\cap A_{t}; d^{'}\precsim d_{i}^{t}}}  \omega ( d^{'} )\\ &\leq & \sum_{t} \nu( A_{t} ) + \varepsilon.\eea
Since $\bigcup_{t\in \N} A_{t}\subseteq \bigcup_{i,t\in \N} W_{i}^{t}$,
\[\nu(\cup A_{t} ) \leq  \sum_{t} \nu( A_{t} ) + \varepsilon\]
for each arbitrary $\varepsilon > 0$ and this completes the proof.
\end{proof}

In spite of what seems to be happening for $\mu^{*}$, the measurable sets in this case includes all the power set  of $X$, $2^{X}$.
\begin{theorem}
Every subset of $X$ is a $\nu$-measurable set.
\end{theorem}
\begin{proof}
Suppose that $E$ is an arbitrary subset of $X$. Then for each $A\subseteq X$ and for $\varepsilon> 0$, there exist a sequence
  $\{W_{i}\}\subseteq \F$ such that
\[  \nu( A ) + \varepsilon \geq \sum_{i} \sum_{{d^{'}\in D\cap A; d^{'}\precsim d_{i}}}  \omega ( d^{'} )\]
 for which  $A\subseteq\cup W_{i}$ and  $W_{i}= W(  d_{i} )$ for all $ i\in \N$.\\
 Put $A_{1}= A\cap E$ and $A_{2}= A\cap E^{c}$. Then $A_{1} , A_{2} \subseteq \cup W_{i}$ and so
\bea \nu( A ) + \varepsilon &\geq & \sum_{i} \sum_{{d^{'}\in D\cap A\cap E; d^{'}\precsim d_{i}}}  \omega ( d^{'} ) +  \sum_{i} \sum_{{d^{'}\in D\cap A \cap E^{c}; d^{'}\precsim d_{i}}}  \omega ( d^{'} )\\ &= & \sum_{i} \sum_{{d^{'}\in D\cap A_{1}; d^{'}\precsim d_{i}}}  \omega ( d^{'} ) +  \sum_{i} \sum_{{d^{'}\in D\cap A_{2}; d^{'}\precsim d_{i}}}  \omega ( d^{'} )\\& \geq &\nu(A_{1}) + \nu(A_{2}).\eea
Since this hold for every $\varepsilon >0$, thus
\[  \nu( A ) \geq  \nu( A \cap E ) +  \nu( A \cap E^{c} ) \ \ \ \forall  A\subseteq X\]
Therefore $E$ is  $\nu$-measurable.
 \end{proof}

\begin{corollary}
$\nu$ is a measure on $2^{X}$.
\end{corollary}

\section{Applications in economics}
In terms of economics, the set $(   X , \succsim )$ with the conditions mentioned in previous sections can be considered as a consumption set. In any model of consumer choice, a consumption set
is the set of all alternatives that the consumer can conceive. In this case, $X$ is considered as a closed convex subset of
$\R ^{n}_{+}$ which contains $0$. The pseudo-ordering relation $ \succsim$ is called the preference relation and the sets
$W(x_{0})$, $B(x_{0})$ and $I(x_{0})$ are called 'worse than' set, 'preferred to' set and 'indifference' set, respectively. One of the most important issues is the existence of a continuous utility function representing a preference relation; although, in many cases, the weaker property 'upper semi-continuity' suffices.
For more information about these economic terms one can refer to \cite{jr}.

In this section we suppose that $\succsim$ is an order separable relation on $X$.
Define the  function $\theta$ from $X$ into  $\R$ in the  form
\[ \theta( x):=  \nu( W( x) ) \ \ \ \forall x\in X\]
 and put  $I\left(D\right)   = \left\lbrace  x\in X; x \thicksim d \  \textrm{for some} \ d \in D \right\rbrace $. In the following,
  we examine some of the properties of $\theta$ as a utility function .
\begin{theorem}\label{t1}
The mapping  $\theta$ is a measurable utility function represents
$\succsim$ on $X$ and is continuous on  $X\setminus I\left( D\right)$ in the $D$-order topology of $X$.
\end{theorem}
\begin{proof}
 Note that for each $d\in D$, $  \theta( d ) = \sum_{d^{'}\in D\cap W(d) }  \omega ( d^{'} )$. Hence for each $x, y \in X$ if $x\sim y$, then $W(x) = W(y)$ and so $\theta(x) = \theta(y)$; if $x \prec y$, there are  $ d_{1} ,d_{2}\in D$ with $x \precsim d_{1} \prec d_{2} \precsim y$ and so
$$ \theta (x) =  \nu( W( x ) ) \leq  \nu( W(d_{1}) ) < \nu( W(d_{2}) ) \leq  \nu( W( y ) ) = \theta (y);$$
that is,  $\theta$  represents $\succsim$ on $X$.
 Now for each  $\varepsilon> 0$ there is $ k\in \N$ such that
\[ \forall m,n\in \N \ \ \ \ \ m > n \geq k \Rightarrow \sum_{i=n+1}^{m}\varepsilon_{i}<\varepsilon.\]
Suppose  $x \in X\setminus I\left( D\right)$  . By order separability of $\succsim$, the sets $W(x) \cap D$ and $B(x) \cap D$ are infinite and for each $ d_{1} \in W(x) \cap D$ and $d_{2}\in B(x) \cap D$ there exist  $ d_{3} ,d_{4}\in D$ such that $ d_{1} \prec d_{3} \prec x \prec d_{4} \prec d_{2}$. Since there are only finitely many $d^{'}\in D$ with  $t(d^{'}) < k  $ we can choose $d_{1}\in W(x) \cap D$ and $d_{2}\in B(x) \cap D$ such that $t(d^{'})\geq k$ for each $d^{'}\in D$ with $ d_{1}\precsim d^{'}\prec d_{2}$ .
Put $V=B(d_{1}) \cap W(d_{2}) = (d_{1} , d_{2})$. Trivially $V$ is an open set in $D$-order topology contains $x$. Let $y\in V$; then
 \bea\vert \theta(y) - \theta(x) \vert &<& \theta(d_{2}) - \theta(d_{1}) \\ &=& \nu( W(d_{2}))-\nu( W(d_{1}))\\ &=& \sum_{d^{'}\in D ; d_{1}\precsim d^{'}\prec d_{2}} \omega ( d^{'} ) \\
 &=&  \sum_{d^{'}\in D ; d_{1}\precsim d^{'}\prec d_{2}} \varepsilon_{t ( d^{'} ) } < \varepsilon.\eea
\end{proof}

In the above theorem, for each  $x \in X\setminus I\left( D\right)$, one can choose $ d_{1} \in W(x) \cap D$
 and $d_{2}\in B(x) \cap D$ so that  $t(d^{'})\geq k$ for each $d^{'}\in D$ with $ d_{1}\prec d^{'}\precsim d_{2}$;
  in this case, in a similar way for the utility function $u$ we have
  \[ \vert u(y) -u(x) \vert < \sum_{d^{'}\in D ; d_{1}\prec d^{'}\precsim d_{2}} \omega ( d^{'} ) < \varepsilon. \]
 for all $ y \in   (d_{1} , d_{2})$. This discussion is summarized in the following theorem.
 \begin{theorem}
The utility function $u$ is  continuous on  $X\setminus I\left( D\right)$ in the $D$-order topology of $X$.
\end{theorem}
As an straightforward consequence of theorem \ref{t1}, It can be said that $\theta$ is continuous on  $X\setminus I\left( D\right)$ in the $D$-lower order topology. In the next theorem, we  discuss  the upper semi-continuity of it.
\begin{theorem}
 The function $\theta$  is upper semi-continuous  in the $D$-lower order topology in

 $(i)$  any point $x$ of $I\left( D\right)$ for which there exists  $d^{'}\in B(x) \cap D$ with $ (x , d^{'}) = \emptyset $;

 $(ii)$   any point of  $X\setminus I\left( D\right)$.
\end{theorem}
\begin{proof}suppose that $r \in \R$ and $\lbrace x\in X; \theta(x) < r \rbrace$ is a non trivial subset of $X$.
If $x \in I(D)$ and there is a $d^{'}\in B(x) \cap D$ with $ (x , d^{'}) = \emptyset $, then
$$\lbrace y \in X; y\precsim x\rbrace = \lbrace  y \in X; y\prec d^{'}\rbrace = W (d^{'})$$ is a neighborhood of $x$ in  $D$-lower order topology and for each $y \in W (d^{'})$,
$ \theta (y) \leq \theta (x) < r$ and this proves $(i)$.

Now let   $x \in X\setminus I(D)$. Then there is   $ k\in \N$ such that for all  $ n \geq k$,  $\sum_{i=n+1}^{\infty}\varepsilon_{i}<r - \theta (x)$.  similar to the proof of the last theorem, there exist
 $ d_{1} \in W(x) \cap D$ and $d_{2}\in B(x) \cap D$ such that $t(d^{'})\geq k$ for all $d^{'}\in D$ with  $d_{1} \precsim d^{'} \prec d_{2}$. For each $y \in W(d_{2})$,
 \bea  \theta (y) < \theta (d_{2})  &=& \sum_{d^{'}\in D \cap W(d_{2})} \omega ( d^{'} ) \\ &=& \sum_{d^{'}\in D ;  d^{'} \prec d_{1}} \omega ( d^{'} )  +
  \sum_{d^{'}\in D ; d_{1}\precsim d^{'}\prec d_{2}} \omega ( d^{'} ) \\ &<& \theta (d_{1}) + ( r -  \theta (x) )
  \\ &<& \theta (x) + ( r -  \theta (x) ) = r; \eea
 that is, $W(d_{2})$ is a neighborhood of $x$ which is a subset of $\lbrace x\in X; \theta(x) < r \rbrace$; this completes the proof of $(ii)$.
\end{proof}

We hope that applying $\theta$ as a measure function will lead to more desirable results about the utility function. For example, If $\lbrace x_{n}\rbrace$ is an increasing sequence in  $X$ (i.e., $ x_{1} \precsim x_{2} \precsim x_{3} \precsim ...$), then $\lbrace W( x_{n} )\rbrace$ is an increasing sequence of measurable sets in $X$  and by the property (continuity from below) of measure,
$$ \lim_{n}\theta(x_{n}) =  \lim_{n}\nu(W(x_{n})) = \nu(\bigcup_{n}W(x_{n})).$$
In this case, if $\lbrace x_{n}\rbrace$ is in $X\setminus I\left( D\right)$ and converges to some element $x_{0}$ of this space, in $D$-order topology, then  by  theorem \ref{t1},  $\theta(x_{0}) = \lim_{n}\theta(x_{n}) = \nu(\bigcup_{n}W(x_{n})).$

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\begin{thebibliography}{99} % Enter references in alphabetical order and according to the following format.
\bibitem{f} G. B. Folland, {\it Real analysis, Modern techniques and their applications: Second edition}, John Wiley and Sons, Inc.  Canada, 1999.

\bibitem{j} J.-Y. Jaffray,{\it Existence of a continuous utility function: An elementary proof}, Econometrica 43 { No. 5/6 }, (1975) 981-983.

\bibitem{jr} G. A. Jehle and P. J. Reny, {\it Advanced microeconomic theory: Third edition}, Prentice Hall, England, 2011.

\bibitem{mk} T. Mitra and M. Kemal Ozbek, {\it On representation of monotone preference orders in a sequence space}, Soc Choice Welf 41, (2013) 473-487.

\bibitem{s} D. Sondermann, {\it utility representations for partial orders}, journal of economic theory 23, (1980) 183-188.

\bibitem{vw} M.Voorneveld and J. W. Weibull, {\it outer measure and utility}, Stockholm School of Economics, Working Paper Series in Economics and Finance { No. 704 }, (2009) 1-8.
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{\small

\noindent{\bf Somayeh Mohammadzadeh}

\noindent Assistant Professor of Mathematics

\noindent Department of Mathematics

\noindent Faculty of  Science

\noindent University of Bojnord, P. O. Box 1339 

\noindent Bojnord , Iran

\noindent E-mail: smohamad@ub.ac.ir}\\



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