Measurable representations of preference orders

Author:
R. Daniel Mauldin

Journal:
Trans. Amer. Math. Soc. **275** (1983), 761-769

MSC:
Primary 90A06; Secondary 04A15, 28C15, 54H05

DOI:
https://doi.org/10.1090/S0002-9947-1983-0682730-4

MathSciNet review:
682730

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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*.

**[1]**J. P. Burgess,*Classical hierarchies from a modern standpoint, Part I*. -*sets*, Fund. Math. (to appear).**[2]**-, Personal communication, 1981.**[3]**-,*From preference to utility, a problem of descriptive set theory*, preprint.**[4]**Douglas Cenzer and R. Daniel Mauldin,*Measurable parametrizations and selections*, Trans. Amer. Math. Soc.**245**(1978), 399–408. MR**511418**, https://doi.org/10.1090/S0002-9947-1978-0511418-3**[5]**G. Debreu,*Continuity properties of Paretian utility*, Internat. Econom. Rev.**5**(1964), 285-293.**[6]**C. Dellacherie,*Un cours sur les ensembles analytiques*, Analytic Sets, edited by C. A. Rogers et al., Academic Press, New York, 1980.**[7]**Arnold M. Faden,*Economics of space and time*, Iowa State University Press, Ames, Iowa, 1977. The measure-theoretic foundations of social science; With a foreword by Martin J. Beckmann. MR**0469204****[8]**D. Fremlin, Personal communication, 1981.**[9]**K. Kuratowski,*Topology. Vol. I*, New edition, revised and augmented. Translated from the French by J. Jaworowski, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe, Warsaw, 1966. MR**0217751****[10]**R. Daniel Mauldin,*The boundedness of the Cantor-Bendixson order of some analytic sets*, Pacific J. Math.**74**(1978), no. 1, 167–177. MR**0474236****[11]**-,*Measurable constructions of preference orders*, unpublished manuscript.**[12]**J. T. Rader,*The existence of a utility function to represent preferences*, Rev. Econom. Stud.**30**(1963), 229-232.**[13]**Jean Saint-Raymond,*Boréliens à coupes 𝐾_{𝜎}*, Bull. Soc. Math. France**104**(1976), no. 4, 389–400. MR**0433418****[14]**Steven E. Shreve,*Probability measures and the 𝐶-sets of Selivanovskij*, Pacific J. Math.**79**(1978), no. 1, 189–196. MR**526678****[15]**Daniel H. Wagner,*Survey of measurable selection theorems*, SIAM J. Control Optimization**15**(1977), no. 5, 859–903. MR**0486391**, https://doi.org/10.1137/0315056**[16]**Eugene Wesley,*Borel preference orders in markets with a continuum of traders*, J. Math. Econom.**3**(1976), no. 2, 155–165. MR**0439054**, https://doi.org/10.1016/0304-4068(76)90024-0

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
90A06,
04A15,
28C15,
54H05

Retrieve articles in all journals with MSC: 90A06, 04A15, 28C15, 54H05

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1983-0682730-4

Keywords:
Preference order,
continuous order preserving map,
universally measurable,
analytic set

Article copyright:
© Copyright 1983
American Mathematical Society