Combinatorial methods in the study of marginal problems over separable spaces
The paper deals with multidimensional (infinite) forms of the following question: When has a triple of separable spaces \(X,Y,Z\) the property that to any Borel probability measure \(P\) on \(X\times Y\) and \(Q\) on \(Y\times Z\) one can finda measure \(R\) on \(X\times Y\times Z\) with the marginals \(P,Q\). In the second part of the paper the question is extended to the case of projective systems of spaces. A basic property of the underlying spaces is the property that each probability measure is tight, which is called 'universally measurable' in this paper. The construction is based on the analysis of the combinatorial structure of the set of marginals (simplicial complexes) as introduced in the case of finite sets by \textit{N. N. Vorob'ev}, Consistent families of measures and their extensions. Teor. Veroyatn. Primen. 7, 153-169 (1962).
scientific article; zbMATH DE number 3828925