On regular extensions of measures

Bachman, George; Sultan, Alan

doi:10.2140/pjm.1980.86.389

Cited by 33 publications

(25 citation statements)

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“…Assuming Martin's axiom MA(ω 1 ), Fremlin [8] showed that if a compact space K admits a measure of uncountable type then K can be continuously mapped onto [0,1] ω 1 , so in particular K must have uncountable tightness. Since P (K) contains a subspace homeomorhic to K it follows that Problem 1.1 has a positive solution under MA(ω 1 ).…”

Section: Problem 11 Suppose That P (K) Has Countable Tightness Doementioning

confidence: 99%

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Countable tightness in the spaces of regular probability measures

Plebanek

Sobota

2015

Fund. Math.

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Abstract. We prove that if K is a compact space and the space P (K × K) of regular probability measures on K × K has countable tightness in its weak * topology, then L 1 (µ) is separable for every µ ∈ P (K). It has been known that such a result is a consequence of Martin's axiom MA(ω 1 ). Our theorem has several consequences; in particular, it generalizes a theorem due to Bourgain and Todorčević on measures on Rosenthal compacta.

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Section: Problem 11 Suppose That P (K) Has Countable Tightness Doementioning

confidence: 99%

“…Talagrand [20] showed that if K admits a measure of type ω 2 then P (K) can be continuously mapped onto [0,1] ω 2 . Thus the following analogue of 1.1 holds true: if τ (P (K)) ≤ ω 1 then every measure µ ∈ P (K) is of type ≤ ω 1 .…”

Section: Problem 11 Suppose That P (K) Has Countable Tightness Doementioning

confidence: 99%

Countable tightness in the spaces of regular probability measures

Plebanek

Sobota

2015

Fund. Math.

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“…Let 3s ç 2X be given and let a be a real number. The condition 1(3) > a is necessary and sufficient for the existence of a probability quasi-measure m on 2X such that m\^> > a. Consequently, 1(3) = supm{^> (3), where the supremum is taken over all probability quasi-measures tp on 2X.…”

Section: Preliminariesmentioning

confidence: 99%

“…There have been developed several techniques of constructing regular extensions of measures; see [10,3,1], and also [7,12] where the problem is studied in the context of group-valued measures (3Í and Sf are usually assumed to be lattices only and cr-additive quasi-measures are discussed). Problems of this type appear quite naturally in topological measure theory; for example, the question which Baire measures have regular Borel extensions is of great interest (cf.…”

Section: On Measure Extension Problemmentioning

confidence: 99%

Families of sets of positive measure

Plebanek¹

1992

Trans. Amer. Math. Soc.

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Abstract.We present a combinatorial description of those families 5a of sets, for which there is a finite measure p. such that inf{ß{P) : P 6 a"} > 0 . This result yields a topological characterization of measure-compactness and Borel measure-compactness. It is also applied to a problem on the existence of regular measure extensions.The main part of this paper deals with the following problem. Given a family ¿P of subsets of a certain set X, under what conditions does there exist a finite measure p. defined on some o-algebra containing £P such that M{p(P) : P e 3°} > 0! The solution to the "finitely additive" version of this problem has been known since Kelley [11] introduced the notion of intersection numbers in order to characterize Boolean algebras having strictly positive (finitely additive) measures. Kelley's idea was subsequently used to describe compact topological spaces that support Radon measures (cf. In §2 of this paper we introduce a combinatorial condition involving intersection numbers which provides an answer to the question mentioned above. The subject is investigated in the context of measures being regular with respect to a given ¿-lattice. This setting is suitable for some applications of the main result. Section 3 contains a characterization of measure-compact and Borel measure-compact topological spaces which seems to answer the problems posed by Wheeler [19] and Gardner-Pfeffer [8]. Next we discuss a problem on the existence of regular measure extensions.

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“…We make use of the following extension theorem a proof of which can be found in [9]. THEOREM 3.8. 2) WA n WA n wn).…”

Section: Definitions and Notationsmentioning

confidence: 99%