On the Grothendieck and Nikodym properties for algebras of Baire, Borel and universally measurable sets

Graves, William H.; Wheeler, Robert F.

doi:10.1216/rmj-1983-13-2-333

Cited by 16 publications

(12 citation statements)

References 25 publications

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“…Recall that while there are many consistent or even ZFC examples of Boolean algebras with the Nikodym property but without the Grothendieck property, see e.g. [31], [21], [37], so far only one example of an algebra with the Grothendieck property and without the Nikodym property has been found-the construction was obtained by Talagrand [38] under the assumption of the Continuum Hypothesis. 6.2.…”

Section: Cardinal Characteristics Of the Continuummentioning

confidence: 99%

Convergence of measures in forcing extensions

Sobota

Zdomskyy

2019

Isr. J. Math.

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We prove that if A is a σ-complete Boolean algebra in a model V of set theory and P ∈ V is a proper forcing with the Laver property preserving the ground model reals non-meager, then every pointwise convergent sequence of measures on A in a P-generic extension V [G] is weakly convergent, i.e. A has the Vitali-Hahn-Saks property in V [G]. This yields a consistent example of a whole class of infinite Boolean algebras with this property and of cardinality strictly smaller than the dominating number d. We also obtain a new consistent situation in which there exists an Efimov space.

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Section: Cardinal Characteristics Of the Continuummentioning

confidence: 99%

Convergence of measures in forcing extensions

Sobota

Zdomskyy

2019

Isr. J. Math.

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“…It is henceforth assumed that S(s/) is endowed with the universal measure topology T, described in [16] and [24] and subsequently studied in [4]- [8] and [17]- [19], so that the measure x is sea and the linear operator O is r-continuous for each <É> which is sea. Let L(s/)be the completion of S(s/) (with topology also denoted r but notationally suppressed), and let <%(s/) be the completion of S(s/) in the sup norm.…”

Section: Vector Measures As Operatorsmentioning

confidence: 99%

“…Let M(£2, X) be the space of all regular ca X-valued Borel measures on £2. Stonean considerations and the isometries above are shown in [4] and [19] to lead to natural isometries…”

Section: $(P) =mentioning

confidence: 99%

Compactness and Weak Compactness in Spaces of Compact-Range Vector Measures

Graves

Ruess

1984

Can. j. math.

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This paper features strong and weak compactness in spaces of vector measures with relatively compact ranges in Banach spaces. Its tools are the measure-operator identification of [16] and [24] and the description of strong and weak compactness in spaces of compact operators in [10], [11], and [29].Given a Banach space X and an algebra of sets, it is shown in [16] that under the usual identification via integration of X-valued bounded additive measures on with X-valued sup norm continuous linear operators on the space of -simple scalar functions, the strongly bounded, countably additive measures correspond exactly to those operators which are continuous for the coarser (locally convex) universal measure topology τ on . It is through the latter identification that the results on strong and weak compactness in [10], [11], and [29] can be applied to X-valued continuous linear operators on the generalized DF space to yield results on strong and weak compactness in spaces of vector measures.

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“…Cantor separable Boolean algebras & are characterized in terms of their Stone space X & by the fact that each non-empty zero set has a non-empty interior. Completely regular spaces X with the aforementioned property are called almost P-spaces in [17] and have been studied in [7], [10] and [27]. Thus, & is Cantor separable if X & is an almost P-space.…”

Section: When Is the Algebra Of Regular Sets For A Finitely Additive mentioning

confidence: 99%

“…fSN \ N is the most familiar example of an almost P-space [9, 65.8]. Graves and Wheeler in [10] give a method for producing a large class of almost P-spaces. The following proposition was pointed out by R. F. Wheeler.…”

Section: When Is the Algebra Of Regular Sets For A Finitely Additive mentioning

confidence: 99%

When is the algebra of regular sets for a finitely additive borel measure a α-algebra?

Armstrong

1982

J Aust Math Soc A

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It is shown that the algebra of regular sets for a finitely additive Borel measure p. on a compact Hausdorff space is a a-algebra only if it includes the Baire algebra and p is countably additive on the o-algebra of regular sets. Any infinite compact Hausdorff space admits a finitely additive Borel measure whose algebra of regular sets is not a o-algebra. Although a finitely additive measure with a a-algebra of regular sets is countably additive on the Baire o-algebra there are examples of finitely additive extensions of countably additive Baire measures whose regular algebra is not a o-algebra. We examine the particular case of extensions of Dirac measures. In this context it is shown that all extensions of a {0, l}-valued countably additive measure from a o-algebra to a larger a-algebra are countably additive if and only if the convex set of these extensions is a finite dimensional simplex.1980 Mathematics subject classification (Amer. Math. Soc): 28 C 15, 28 A 60, 54 G 10.

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On the Grothendieck and Nikodym properties for algebras of Baire, Borel and universally measurable sets

Cited by 16 publications

References 25 publications

Convergence of measures in forcing extensions

Convergence of measures in forcing extensions

Compactness and Weak Compactness in Spaces of Compact-Range Vector Measures

When is the algebra of regular sets for a finitely additive borel measure a α-algebra?

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