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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Duality and perfect probability spaces


Authors: D. Ramachandran and L. Rüschendorf
Journal: Proc. Amer. Math. Soc. 124 (1996), 2223-2228
MSC (1991): Primary 60A10, 28A35
MathSciNet review: 1342043
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Abstract: Given probability spaces $(X_i,\mathcal {A}_i,P_i), i=1,2,$ let $\mathcal {M}(P_1,P_2)$ denote the set of all probabilities on the product space with marginals $P_1$ and $P_2$ and let $h$ be a measurable function on $(X_1 \times X_2,\mathcal {A}_1 \otimes \mathcal {A}_2).$ Continuous versions of linear programming stemming from the works of Monge (1781) and Kantorovich-Rubin\v{s}tein (1958) for the case of compact metric spaces are concerned with the validity of the duality

\begin{align*}&\sup \{ \int h\ dP: P \in \mathcal {M}(P_1,P_2) \} \\ % &\qquad = \: \inf \{ \sum _{i=1}^{2}\ \int h_i\ dP_i : h_i \in \mathcal {L}^1 (P_i) \; \; and \; \; h \leq {\oplus }_i\ h_i\} \end{align*}

(where $\mathcal {M}(P_1,P_2)$ is the collection of all probability measures on $(X_1 \times X_2,\mathcal {A}_1 \otimes \mathcal {A}_2)$ with $P_1$ and $P_2$ as the marginals). A recently established general duality theorem asserts the validity of the above duality whenever at least one of the marginals is a perfect probability space. We pursue the converse direction to examine the interplay between the notions of duality and perfectness and obtain a new characterization of perfect probability spaces.


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Additional Information

D. Ramachandran
Affiliation: Department Of Mathematics and Statistics, California State University, 6000 J Street, Sacramento, California 95819-6051
Email: chandra@csus.edu

L. Rüschendorf
Affiliation: California State University, Sacramento and Universität Freiburg
Address at time of publication: Institut für Mathematische Stochastik, Albert-Ludwigs-Universität, Hebelstr. 27, D-79104 Freiburg, Germany
Email: ruschen@buffon.mathematik.uni-freiburg.de

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03462-4
PII: S 0002-9939(96)03462-4
Keywords: Duality theorem, marginals, perfect measure, Marczewski function
Received by editor(s): December 15, 1994
Additional Notes: Research supported in part by an Internal Awards Grant from the California State University, Sacramento
Communicated by: Richard T. Durrett
Article copyright: © Copyright 1996 American Mathematical Society