On the extremality of measure extensions. (English) Zbl 0663.28004

Bierlein, D.; Stich, W. J. A.

Manuscr. Math. 63, No. 1, 89-97 (1989).

Main result: Let \(\mu\) denote a finite measure on a \(\sigma\)-algebra \({\mathfrak A}\) of subsets of a set \(\Omega\) and introduce \({\mathfrak A}'\) as the \(\sigma\)-algebra of subsets of \(\Omega\) generated by \({\mathfrak A}\) and a finite decomposition \(A_ 1,A_ 2,...,A_ m\) of \(\Omega\). Then any extension \(\mu\) ’ of \(\mu\) to \({\mathfrak A}'\) as a measure can be represented as a mixture of extremal extension of \(\mu\) to \({\mathfrak A}'\), where the mixing probability measure is discrete. It might be interesting to mention, that the existence of a mixing probability measure living on the set of extremal extensions of \(\mu\) to \(\mu\) ’ as a measure can also be proved by means of Choquet’s theorem under the additional assumption, that \({\mathfrak A}'\) is countably generated. However, the discreteness of the representing probability measure does not follow from Choquet’s theorem. Nevertheless, Choquet’s theorem admits a finitely additive version, if \({\mathfrak A}\) is replaced by a countable algebra of subsets of \(\Omega\) and \({\mathfrak A}'\) denotes the algebra of subsets of \(\Omega\) generated by \({\mathfrak A}\) and countable many subsets of \(\Omega\).

Reviewer: D.Plachky

MSC:

28A33

Spaces of measures, convergence of measures

Keywords:

extremal extension of measures; mixing probability measure

References:

[1]	Bierlein, D.: Über die Fortsetzung von Wahrscheinlichkeitsfeldern. Z. Wahrscheinlichkeitstheorie verw. Gebiete 1, 28-46 (1962) · Zbl 0109.10601 · doi:10.1007/BF00531770
[2]	Bierlein, D.: Measure extensions and measurable neighbours of a function. Lect. Notes Math. 794, 1-23 (1980) · Zbl 0425.28003 · doi:10.1007/BFb0088207
[3]	Douglas, R. G.: On extremal measures and subspace density. Mich. Math. J. 11, 243-246 (1964) · Zbl 0121.33102 · doi:10.1307/mmj/1028999137
[4]	Ershov, M. P.: The Choquet theorem and stochastic equations. Analysis Mathematica 1, 259-271 (1975) · Zbl 0322.60004 · doi:10.1007/BF02333176
[5]	Ershov, M. P.: Second disintegration of measures. Institutsbericht No. 135, Math. Inst. Univ. Linz (1979) · Zbl 0432.60061
[6]	Plachky, D.: Zur Fortsetzung additiver Mengenfunktionen. Habilitationsschrift, Universität Münster (1971)
[7]	Plachky, D.: Extremal and monogenic additive set functions. Proc. Am. Math. Soc. 54, 193-196 (1976) · Zbl 0285.28005 · doi:10.1090/S0002-9939-1976-0419711-3
[8]	von Weizsäcker, H.: Der Satz von Choquet-Bishop-de Leeuw für konvexe nicht kompakte Mengen straffer Maße über beliebigen Grundräumen. Math. Z. 142, 161-165. (1975) · Zbl 0287.28002 · doi:10.1007/BF01214946
[9]	von Weizsäcker, H. and Winkler, G.: Integral representations in the set of solutions of a generalized moment problem. Math. Ann. 246, 23-32 (1979) · doi:10.1007/BF01352023
[10]	Winkler, G.: On the integral representation in convex noncompact sets of tight measures. Math. Z. 158, 71-77 (1978) · doi:10.1007/BF01214567