1962/01/01 by N. N. Vorob’ev, N. N. Vorob’ëv · 4 citations
Mathematics · #Advanced Topology and Set Theory
paper · doi:10.1137/1107014
crossref issued 1962/01/01 · crossref published 1962/01/01 · crossref published-print 1962/01/01 · openalex publication_date 1962/01/01 · crossref created 2005/03/07 · crossref deposited 2017/01/29 · openalex created_date 2025/10/10 · crossref indexed 2026/07/28 · openalex updated_date 2026/07/28
Let Σ be a family of Borel fields of subsets of a set S and μ_\mathfrakS probabilistic measures on measurable spaces ⟨ \mathfrakS,S ⟩ , where \mathfrakS ∈ Σ . The family of measures μ_\mathfrakS , \mathfrakS ∈ Σ is denoted by μΣ . The measures μ_\mathfrakS1 and μ_\mathfrakS2 are said to be consistent if μ_\mathfrakS1 (A) = μ_\mathfrakS2 (A) for any A ∈ \mathfrakS1 ∩ \mathfrakS2 . If any pair of measures of the family μΣ is consistent, the family itself is referred to as consistent. The consistent family μΣ is said to be extendable if there is a measure μ[Σ ] on the measurable space ⟨ [Σ ],S ⟩ consistent with each measure of μΣ ([Σ ] is the smallest Borel field containing all \mathfrakS ∈ Σ ). For the purposes of the theory of games the following special case of extendability is important. Let \bf \mathfrakK be a finite complete complex and M the set of its vertices. Let a finite set Sa correspond to each vertex a of \bf \mathfrakK and the set SA = Π α ∈ A Sα to each subset A ⊂ M. Let \mathfrakSK = \ XK :XK = YK × SM - K , YK ⊂ SK \, K ∈ \bf \mathfrakK;μ K is a measure on ⟨ \mathfrakSK ,SM ⟩ and μ _\bf \mathfrakK is the family of all such measures. The extendability of the family μ _\bf \mathfrakK is closely related with the combinatorial properties of the complex \bf \mathfrakK . Any maximal face of the complex \bf \mathfrakK is said to be an extreme face if it has proper vertices (i.e. such vertices which do not belong to any other maximal face of \bf \mathfrakK ). If T is an extreme face of \bf \mathfrakK the complex \bf \mathfrakK^* obtained by removing from \bf \mathfrakK all proper vertices of T with their stars is said to be a normal subcomplex of \bf \mathfrakK . A complex \bf \mathfrakK is said to be regular if there is a sequence \bf \mathfrakK = \bf \mathfrakK0 ⊃ \bf \mathfrakK1 ⊃ ⋯ ⊃ \bf \mathfrakKn of subcomplexes of \bf \mathfrakK where \bf \mathfrakKi is a normal subcomplex of \bf \mathfrakKi - 1 ,i = 1, ⋯ ,n, and the last member vanishes. The main results of the paper consists in the following statement. Theorem. The regularity of the complex\bf \mathfrakKis a necessary and sufficient condition of extendability of any consistent family ofμ_\bf \mathfrakKof measures.