Measurable representations of preference orders
Author:
R. Daniel Mauldin
Journal:
Trans. Amer. Math. Soc. 275 (1983), 761769
MSC:
Primary 90A06; Secondary 04A15, 28C15, 54H05
MathSciNet review:
682730
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Abstract: A continuous preference order on a topological space is a binary relation which is reflexive, transitive and complete and such that for each and are closed. Let and be complete separable metric spaces. For each in , let be a nonempty subset of , let be a continuous preference order on and suppose is a Borel set. Let . Theorem 1. There is an measurable map from into so that for each is a continuous map of into and if and only if . (Here forms the sets of Selivanovskii and is a Borel field on .) Theorem 2. If for each is a compact subset of , then the map 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 be the map constructed in Theorem 1. If is a probability measure defined on the Borel subsets of , then there is a Borel set such that and such that the restriction of to is Borel measurable.
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 J. P. Burgess, Classical hierarchies from a modern standpoint, Part I. sets, Fund. Math. (to appear).
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 , Personal communication, 1981.
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 , From preference to utility, a problem of descriptive set theory, preprint.
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 D. Cenzer and R. D. Mauldin, Measurable parametrizations and selections, Trans. Amer. Math. Soc. 245 (1978), 399408. MR 511418 (80i:28010)
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 G. Debreu, Continuity properties of Paretian utility, Internat. Econom. Rev. 5 (1964), 285293.
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 C. Dellacherie, Un cours sur les ensembles analytiques, Analytic Sets, edited by C. A. Rogers et al., Academic Press, New York, 1980.
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 A. M. Faden, Economies of space and time, The measuretheoretic foundations of social science, Iowa State Univ. Press, Ames, Iowa, 1977. MR 0469204 (57:8998)
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 D. Fremlin, Personal communication, 1981.
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 R. D. Mauldin, The boundedness of the CantorBendixson order of some analytic sets, Pacific J. Math. 74 (1978), 167177. MR 0474236 (57:13883)
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 , Measurable constructions of preference orders, unpublished manuscript.
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 J. T. Rader, The existence of a utility function to represent preferences, Rev. Econom. Stud. 30 (1963), 229232.
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 J. SaintRaymond, Boréliens à coupes , Bull. Soc. Math. France 104 (1976), 389400. MR 0433418 (55:6394)
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 S. E. Shreve, Probability measures and the sets of Selivanovskii, Pacific J. Math. 79 (1978), 189196. MR 526678 (80d:28008)
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 D. Wagner, Survey of measurable selection theorems, SIAM J. Control Optimization 15 (1977), 859903. MR 0486391 (58:6137)
 [16]
 E. Wesley, Borel preference orders in markets with a continuum of traders, J. Math. Econom. 3 (1976), 155165. MR # 11955 MR 0439054 (55:11955)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198306827304
PII:
S 00029947(1983)06827304
Keywords:
Preference order,
continuous order preserving map,
universally measurable,
analytic set
Article copyright:
© Copyright 1983
American Mathematical Society
