A continuous generalization of the transversal property

Extract

A transversal for a set-system is a one-to-one choice function. A necessary and sufficient condition for the existence of a transversal in the case of finite sets was given by P. Hall (see [4, 3]). The corresponding condition for the case when countably many countable sets are given was conjectured by Nash-Williams and later proved by Damerell and Milner [2]. B. Bollobás and N. Varopoulos stated and proved the following measure theoretic counterpart of Hall's theorem: if (X, μ) is an atomless measure space, ℋ = {H_i: i∈I} is a family of measurable sets with finite measure, λ_i (i∈I) are non-negative real numbers, then we can choose a subset T_i ⊆ H_i with μ(T_i) = λ_i and μ(T_i ∩ T_i′) = 0 (i ≠ i′) if and only if μ({U H_i: iεJ}) ≥ Σ{λ_i: iεJ}: for every finite subset J of I. In this note we generalize this result giving a necessary and sufficient condition for the case when I is countable and X is the union of countably many sets of finite measure.