## On extensions of measures which are maximal with respect to a chain.(English)Zbl 0646.28003

Abstract analysis, Proc. 14th Winter Sch., Srnî/Czech. 1986, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 14, 407-409 (1987).
[For the entire collection see Zbl 0627.00012.]
Main result: Let $$\mu$$ denote a probability measure on a $$\sigma$$-algebra $${\mathfrak A}$$ of subsets of a set X and $${\mathfrak Z}^ a$$chain of subsets of X satisfying
(i) the Lebesgue measure of the closure of the set $$\{\mu$$ *(Z): $$Z\in {\mathfrak Z}\}$$ is zero,
(ii) $$\mu$$ *($$\cap {\mathfrak Z}')=\inf \{\mu$$ *(Z): $$Z\in {\mathfrak Z}'\}$$ is valid for every countable subset $${\mathfrak Z}'$$ of $${\mathfrak Z},$$
then there exists a probability measure $$\nu$$ on the $$\sigma$$-algebra of subsets of X generated by $${\mathfrak A}$$ and $${\mathfrak Z}$$ with the property $$\nu (Z)=\mu$$ *(Z) for every $$Z\in {\mathfrak Z}.$$
An example shows, that the assumption (ii) without condition (i) is not sufficient.
Reviewer: D.Plachky

### MSC:

 28A60 Measures on Boolean rings, measure algebras 28B10 Group- or semigroup-valued set functions, measures and integrals

### Keywords:

maximal extension; extension of measures

Zbl 0627.00012
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