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Duality and perfect probability spaces. (English) Zbl 0863.60005

Summary: 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 G. Monge [Mem. Math. Phys. Acad. Roy. Sci. Paris, 666-704 (1781)] and L. V. Kantorovich and G. S. Rubinshtejn [Vestnik Leningr. Univ. 13, No. 7 (Ser. Mat. Mekh. Astron. No. 2), 52-59 (1958; Zbl 0082.11001)] for the case of compact metric spaces are concerned with the validity of the duality \[ \sup\left\{\int h dP: P\in {\mathcal M}(P_1,P_2)\right\}=\inf\left\{\sum_{i=1}^2 \int h_i dP_i: h_i\in {\mathcal L}^1(P_i)\text{ and }h\leq \oplus_i h_i\right\} \] (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.

MSC:

60A10 Probabilistic measure theory
28A35 Measures and integrals in product spaces

Citations:

Zbl 0082.11001
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References:

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