Strictly positive measures on Boolean algebras

Džamonja, Mirna; Plebanek, Grzegorz

doi:10.2178/jsl/1230396929

Cited by 18 publications

(16 citation statements)

References 25 publications

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“…It follows that P (L) is separable (see e.g. [7]), so when we transfer μ to a measure v on P (L) via the mapping L t → δ t ∈ P (L), the support of v is not separable though v lives on a separable space P (L).…”

Section: Corollary 32 If K Is Rosenthal Compact and Kmentioning

confidence: 99%

On Corson compacta and embeddings of 𝐶(𝐾) spaces

Marciszewski

Plebanek²

2010

Proc. Amer. Math. Soc.

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Section: Corollary 32 If K Is Rosenthal Compact and Kmentioning

confidence: 99%

On Corson compacta and embeddings of 𝐶(𝐾) spaces

Marciszewski

Plebanek²

2010

Proc. Amer. Math. Soc.

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“…Small σ-n-linked spaces supporting no measure. In [DP08] the authors claimed to prove that the space constructed by Todorčević does not support a measure. They were mainly interested in the corollary saying that there is a Boolean algebra of size add(N ) which does not support a measure.…”

Section: 3mentioning

confidence: 99%

Measures and slaloms

Borodulin–Nadzieja

Inamdar

2017

Fund. Math.

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We examine measure-theoretic properties of spaces constructed using the technique of Todorčević from [Tod00, Theorem 8.4]. We show that the existence of strictly positive measures on such spaces depends on combinatorial properties of certain families of slaloms. As a corollary we get that if add(N ) = non(M) then there is a non-separable space which supports a measure and which cannot be mapped continuously onto [0, 1] ω1 . Also, without any additional axioms we prove that there is a non-separable growth of ω supporting a measure and that there is a compactification L of ω with growth of such properties and such that the natural copy of c 0 is complemented in C(L). Finally, we discuss examples of spaces not supporting measures but satisfying quite strong chain conditions. Our main tool is a characterization due to Kamburelis ([Kam89]) of Boolean algebras supporting measures in terms of their chain conditions in generic extensions by a measure algebra.

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“…Indeed, the Stone space of the measure algebra of the Lebesgue measure on [0, 1] satisfies (b) but not (a), while Talagrand [18] constructed under CH two examples showing that (c)⇒(b) and (e)⇒(c) do not hold. Recently, Džamonja and Plebanek [2] gave a ZFC counterexample to (c)⇒(b). In this paper we provide a ZFC example of a compact space K witnessing that the implication (e)⇒(c) does not hold, i.e.…”

Section: It Is Known That the Following Implications Hold (A) =⇒ (B) mentioning

confidence: 99%