On the splicing of measures. (English) Zbl 0549.28004

Let X be a non-empty set, \({\mathcal A}\), \({\mathcal B}\) two \(\sigma\) -algebras on X, \({\mathcal C}_ 0\) the algebra generated by their union, \({\mathcal A}\bigvee {\mathcal B}\) the \(\sigma\) -algebra generated by \({\mathcal C}_ 0\); \(\mu\), \(\nu\) probability measures on \({\mathcal A}\), \({\mathcal B}\), respectively. A splicing of \(\mu\) and \(\nu\) is a measure \(\eta\) on \({\mathcal A}\bigvee {\mathcal B}\) which satisfies \((1)\quad\eta (A\cap B)=\mu (A)\nu (B)\) for all \(A\in {\mathcal A}\), \(B\in {\mathcal B}\). Two conditions, each of which is obviously necessary for the existence of \(\eta\), are \[ (2)\quad\forall A\in {\mathcal A},\quad B\in {\mathcal B}:\quad A\cap B=\emptyset\Rightarrow \mu (A)\nu (B)=0. \]
\[ (3)\quad\forall (A_ n)^{\infty}_{n=1}\subset {\mathcal A},\quad (B_ n)^{\infty}_{n=1}\subset {\mathcal B}:\cup^{\infty}_{n=1}A_ n\cap B_ n=X\Rightarrow\sum^{\infty}_{n=1}\mu (A_ n)\nu (B_ n)\geq 1. \] In 1951 E. Marczewski [Fundam. Math. 38, 217-229 (1951; Zbl 0045.023)] showed that (2) implies the existence of a finitely additive set function \(\eta\) on \({\mathcal C}_ 0\) satisfying (1) and in 1976 D. Stroock [Colloq. Math. 35, 7-13 (1976; Zbl 0337.28001)] showed that (3) implies the existence of a splicing. The authors give new, short and elementary proofs of these results. The idea for their proof of Marczewski’s result is to note that (3)\(\Rightarrow (2)\), and (2) enables the product measure \(\mu\times\nu \) to be projected onto the diagonal D in \(X\times X\) to give a finitely additive set function \(\beta\) via \(\beta (S\cap D):=(\mu\times \nu)(S),\) for \(S\in {\mathcal S}\) the algebra generated by the products \(A\times B\). The Boolean isomorphism of \({\mathcal C}_ 0\) onto \({\mathcal S}\cap D\) given by \(A\cap B\to (A\times B)\cap D\) carries \(\beta\) over to the desired \(\eta\). The authors point out that in an example constructed by Stroock of measures \(\mu\) and \(\nu\) on \(X=[0,1]\) for which (2) holds but (3) fails, the measures \(\mu\) and \(\nu\) are even compact. In the other direction they construct an example in which \(\mu\) and \(\nu\) are compact, (3) holds and no splicing \(\eta\) of \(\mu\) and \(\nu\) is compact. In this example X is a subset of \([0,1]\times [0,1]\) which meets each closed set of positive two-dimensional Lebesgue measure at least once and each vertical and each horizontal line exactly once. [In line 5, p 822 a certain set featuring in this construction is said to have cardinality c because it is an uncountable Borel set. A more elementary justification of this claim is that, due to Fubini, the set happens to have positive Lebesgue measure.]
Reviewer: R.Burckel


28A12 Contents, measures, outer measures, capacities
28A35 Measures and integrals in product spaces
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