Measurable representations of preference orders
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 by R. Daniel Mauldin PDF
 Trans. Amer. Math. Soc. 275 (1983), 761769 Request permission
Abstract:
A continuous preference order on a topological space $Y$ is a binary relation $\preccurlyeq$ which is reflexive, transitive and complete and such that for each $x,\{y:x \preccurlyeq y\}$ and $\{y:y \preccurlyeq x\}$ are closed. Let $T$ and $X$ be complete separable metric spaces. For each $t$ in $T$, let ${B_t}$ be a nonempty subset of $X$, let ${ \preccurlyeq _t}$ be a continuous preference order on ${B_t}$ and suppose $E = \{(t,x,y): x{ \preccurlyeq _t}y\}$ is a Borel set. Let $B = \{(t,x):x \in {B_t}\}$. Theorem 1. There is an $\mathcal {S}(T) \otimes \mathcal {B}(X)$measurable map $g$ from $B$ into $R$ so that for each $t,g(t,\cdot )$ is a continuous map of ${B_t}$ into $R$ and $g(t,x) \leqslant g(t,y)$ if and only if $x{ \preccurlyeq _t}y$. (Here $\mathcal {S}(T)$ forms the $C$sets of Selivanovskii and $\mathcal {B}(X)$ is a Borel field on $X$.) Theorem 2. If for each $t,{B_t}$ is a $\sigma$compact subset of $Y$, then the map $g$ of the preceding theorem may be chosen to be Borel measurable. The following improvement of a theorem of Wesley is proved using classical methods. Theorem 3. Let $g$ be the map constructed in Theorem 1. If $\mu$ is a probability measure defined on the Borel subsets of $T$, then there is a Borel set $N$ such that $\mu (N) = 0$ and such that the restriction of $g$ to $B \cap ((T  N) \times X)$ is Borel measurable.References

J. P. Burgess, Classical hierarchies from a modern standpoint, Part I. $C$sets, Fund. Math. (to appear).
—, Personal communication, 1981.
—, From preference to utility, a problem of descriptive set theory, preprint.
 Douglas Cenzer and R. Daniel Mauldin, Measurable parametrizations and selections, Trans. Amer. Math. Soc. 245 (1978), 399–408. MR 511418, DOI 10.1090/S00029947197805114183 G. Debreu, Continuity properties of Paretian utility, Internat. Econom. Rev. 5 (1964), 285293. C. Dellacherie, Un cours sur les ensembles analytiques, Analytic Sets, edited by C. A. Rogers et al., Academic Press, New York, 1980.
 Arnold M. Faden, Economics of space and time, Iowa State University Press, Ames, Iowa, 1977. The measuretheoretic foundations of social science; With a foreword by Martin J. Beckmann. MR 0469204 D. Fremlin, Personal communication, 1981.
 K. Kuratowski, Topology. Vol. I, Academic Press, New YorkLondon; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751
 R. Daniel Mauldin, The boundedness of the CantorBendixson order of some analytic sets, Pacific J. Math. 74 (1978), no. 1, 167–177. MR 474236 —, Measurable constructions of preference orders, unpublished manuscript. J. T. Rader, The existence of a utility function to represent preferences, Rev. Econom. Stud. 30 (1963), 229232.
 Jean SaintRaymond, Boréliens à coupes $K_{\sigma }$, Bull. Soc. Math. France 104 (1976), no. 4, 389–400. MR 433418
 Steven E. Shreve, Probability measures and the $C$sets of Selivanovskij, Pacific J. Math. 79 (1978), no. 1, 189–196. MR 526678
 Daniel H. Wagner, Survey of measurable selection theorems, SIAM J. Control Optim. 15 (1977), no. 5, 859–903. MR 486391, DOI 10.1137/0315056
 Eugene Wesley, Borel preference orders in markets with a continuum of traders, J. Math. Econom. 3 (1976), no. 2, 155–165. MR 439054, DOI 10.1016/03044068(76)900240
Additional Information
 © Copyright 1983 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 275 (1983), 761769
 MSC: Primary 90A06; Secondary 04A15, 28C15, 54H05
 DOI: https://doi.org/10.1090/S00029947198306827304
 MathSciNet review: 682730