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

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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*.

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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