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Skorohod's representation theorem for sets of probabilities


Authors: Martin Dumav and Maxwell B. Stinchcombe
Journal: Proc. Amer. Math. Soc. 144 (2016), 3123-3133
MSC (2010): Primary 60B10, 60F99, 91B06
DOI: https://doi.org/10.1090/proc/12932
Published electronically: November 20, 2015
MathSciNet review: 3487242
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Abstract: We characterize sets of probabilities, $ \boldsymbol {\Pi }$, on a measure space
$ (\Omega ,\mathcal {F})$, with the following representation property: for every measurable set of Borel probabilities, $ A$, on a complete separable metric space, $ (M,d)$, there exists a measurable $ X:\Omega \rightarrow M$ with $ A = \{X(P): P \in \boldsymbol {\Pi }\}$. If $ \boldsymbol {\Pi }$ has this representation property, then: if $ K_n \rightarrow K_0$ is a sequence of compact sets of probabilities on $ M$, there exists a sequence of measurable functions, $ X_n:\Omega \rightarrow M$ such that $ X_n(\boldsymbol {\Pi }) \equiv K_n$ and for all $ P \in \boldsymbol {\Pi }$, $ P(\{\omega : X_n(\omega ) \rightarrow X_0(\omega )\}) = 1$; if the $ K_n$ are convex as well as compact, there exists a jointly measurable $ (K,\omega ) \mapsto H(K,\omega )$ such that $ H(K_n,\boldsymbol {\Pi }) \equiv K_n$ and for all $ P \in \boldsymbol {\Pi }$, $ P(\{\omega : H(K_n,\omega ) \rightarrow H(K_0,\omega )\}) = 1$.


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Additional Information

Martin Dumav
Affiliation: Department of Economics, Universidad Carlos III de Madrid, Av. de la Universidad, 30, 28911 Leganés, Madrid, Spain
Email: mdumav@gmail.com

Maxwell B. Stinchcombe
Affiliation: Department of Economics, University of Texas, Austin, Texas 78712-0301
Email: max.stinchcombe@gmail.com

DOI: https://doi.org/10.1090/proc/12932
Keywords: Skorohod's representation theorem, strongly zero-one sets of probabilities, multiple prior models of choice
Received by editor(s): May 18, 2012
Received by editor(s) in revised form: July 16, 2014, January 27, 2015, and August 19, 2015
Published electronically: November 20, 2015
Communicated by: David Asher Levin
Article copyright: © Copyright 2015 American Mathematical Society