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

Plebanek, Grzegorz

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

Citations:

Zbl 0627.00012