Measures admitting extremal extensions. (English) Zbl 0612.28002

Hackenbroch, Wolfgang

Arch. Math. 49, 257-266 (1987).

For \({\mathfrak A}\subset {\mathfrak B}\) two \(\sigma\)-algebras of subsets of some set \(\Omega\) and \(\mu\) a finite positive measure on \({\mathfrak A}\) we investigate the convex set \(M({\mathfrak B};\mu)\) of measures \(\nu\) on \({\mathfrak B}\) extending \(\mu\). If \(X\subset \Omega\) is an \({\mathfrak A}\)-atom for \({\mathfrak B}\) of full \(\mu\)-measure (i.e \(X\cap {\mathfrak A}=X\cap {\mathfrak B}\) and \(\mu _ *(\complement X)=0),\) it is shown that \(\nu: \nu (B)=\mu ^ *(X\cap B)\) defines an extremal point of \(M({\mathfrak B};\mu.)\) (This is an obvious generalization of a Dirac measure \(\delta _ a\), viz. \(\delta _ a=\mu ^ *(\{a\}\cap \cdot)\) for \(\mu\) the trivial probability on the trivial \(\sigma\)-algebra \({\mathfrak A}=\{\emptyset,\Omega \}.)\) If \({\mathfrak B}\) is a countable extension of \({\mathfrak A}\), each \(\nu \in ex M({\mathfrak B};\mu)\) arises in this way, but in general this is not true. On the other hand it is shown that, in a certain ”dilation” of the measure space, as constructed in the author’s paper in Arch. Math. 43, 434-439 (1984; Zbl 0538.28002), each extremal extension of \(\mu\) is of the above form. By this description it is proved that the set of measures \(\mu\) on \({\mathfrak A}\) admitting extremal extensions generates a band in the vector lattice \(M({\mathfrak A})-M({\mathfrak A})\) of signed measures. \({\mathfrak A}\)-atoms are also used to construct extremal extensions of a given \(\mu\) on \({\mathfrak A}\) to \({\mathfrak B}\) via extremal extensions to intermediate \(\sigma\)-algebras \({\mathfrak B}':\) \({\mathfrak A}\subset {\mathfrak B}'\subset {\mathfrak B}\).

MSC:

28A12

Contents, measures, outer measures, capacities

Keywords:

generalization of a Dirac measure; dilation; extremal extensions; signed measures

Citations:

Zbl 0538.28002

References:

[1]	A. Ascherl andJ. Lehn, Two principles for extending probability measures. Manuscripta Math.21, 43-50 (1977). · Zbl 0368.60005 · doi:10.1007/BF01176900
[2]	D. Bierlein, Über die Fortsetzung von Wahrscheinlichkeitsfeldern. Z. Wahrsch. Verw. Gebiete1, 28-46 (1962). · Zbl 0109.10601 · doi:10.1007/BF00531770
[3]	R. G. Douglas, On extremal measures and subspace density. Michigan Math. J.11, 243-246 (1964). · Zbl 0121.33102 · doi:10.1307/mmj/1028999137
[4]	M. P. Ersov, Second disintegration of measures. Institutsbericht Nr.135, Math. Inst. Linz, Austria (1979).
[5]	W. Hackenbroch, Dilated sections and extremal preimage measures. Arch. Math.43, 434-439 (1984). · Zbl 0538.28002 · doi:10.1007/BF01193852
[6]	A. Hanen etJ. Neveu, Atomes conditionnels d’un espace de probabilité. Acta Math. Acad. Sci. Hung.17, 443-449 (1966). · Zbl 0144.39601 · doi:10.1007/BF01894889
[7]	J. Lembcke, On simultaneous preimage measures on Hausdorff spaces. LNM945, 112-115, Berlin-Heidelberg-New York 1982. · Zbl 0495.28005
[8]	Z. Lipecki andD. Plachky, On monogenic operators and measures. Proc. Amer. Math. Soc.82, 216-220 (1981). · Zbl 0474.28001 · doi:10.1090/S0002-9939-1981-0609654-6
[9]	J. Neveu, Atomes conditionnels d’espaces de probabilité et théorie de l’information. LNM31, 256-271 Berlin-Heidelberg-New York 1967.
[10]	D. Plachky, Extremal and monogenic additive set functions. Proc. Amer. Math. Soc.54, 193-196 (1976). · Zbl 0285.28005 · doi:10.1090/S0002-9939-1976-0419711-3
[11]	W.Stich, Maßfortsetzungen und Integration nichtmeßbarer Funktionen. Dissertation, Regensburg 1983. · Zbl 0536.28004
[12]	W. Stich, Integralwertmengen bei Maßfortsetzungen. Arch. Math.37, 523-527 (1981). · Zbl 0477.28001 · doi:10.1007/BF01234390