\documentclass[11pt,twoside]{article} \usepackage{amsmath, amsthm, amscd, amsfonts, amssymb, graphicx, color} \usepackage[bookmarksnumbered, colorlinks]{hyperref} \usepackage{float} \usepackage{lipsum} \usepackage{afterpage} \usepackage[labelfont=bf]{caption} \usepackage[nottoc,notlof,notlot]{tocbibind} %\renewcommand\bibname{References} \def\bibname{\Large \bf References} \usepackage{lipsum} \usepackage{fancyhdr} \pagestyle{fancy} \fancyhf{} \renewcommand{\headrulewidth}{0pt} \fancyhead[LE,RO]{\thepage} \thispagestyle{empty} %\afterpage{\lhead{new value}} \fancyhead[CE]{S. Mohammadzadeh} \fancyhead[CO]{Specific Complete Measure in the Structure of a Utility Function} %\topmargin=-1.6cm \textheight 17.5cm% \textwidth 12cm % \topmargin 8mm % \oddsidemargin 20mm % \evensidemargin 20mm % \footskip=24pt % \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{xca}[theorem]{Exercise} %\theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \renewenvironment{proof}{{\bfseries \noindent Proof.}}{~~~~$\square$} \makeatletter \def\th@newremark{\th@remark\thm@headfont{\bfseries}} \makeatletter \newcommand{\M}{\mathcal{M}} \newcommand{\R}{\mathbb{R}} \newcommand{\F}{\mathrm{F}} \newcommand{\N}{\mathbb{N}} \newcommand{\bea}{\begin{eqnarray*}} \newcommand{\eea}{\end{eqnarray*}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % If you want to insert other packages. Insert them here %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\long\def\symbolfootnote[#1]#2{\begingroup% %\def\thefootnote{\fnsymbol{footnote}}\footnote[#1]{#2}\endgroup} \def \thesection{\arabic{section}} \begin{document} %\baselineskip 9mm %\setcounter{page}{} \thispagestyle{plain} {\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} \vspace{4mm} {\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}