## 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

### MSC:

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

### Citations:

Zbl 0045.023; Zbl 0337.28001
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