Convergence of measures and cardinal characteristics of the continuum

Speaker(s)

Damian Sobota

Affiliation

Kurt Gödel Research Center for Mathematical Logic

Date

Dec. 20, 2017, 4:15 p.m.

Room

room 5050

Seminar

Topology and Set Theory Seminar

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Let A be a Boolean algebra. We say that A has the Nikodym property if every pointwise convergent sequence of measures on A is weak* convergent. Similarly, A has the Grothendieck property if every weak* convergent sequence of measures on A is weakly convergent (equivalently: convergent on every Borel subset of the Stone space of A). The Nikodym property is strictly related to barrelled topological vector spaces (and hence the classical Banach-Steinhaus theorem -- the Uniform Boundedness Principle), while the Grothendieck property -- to the issues of complementability of the Banach space c_0 in the C(K)-spaces. During my talk I will look at the properties from the following set-theoretic point of view. Let nik and gr denote the minimal cardinality of an infinite Boolean algebra with the Nikodym property and the Grothendieck property, respectively. I will show that both nik and gr are cardinal characteristics of the continuum, I will provide lower and upper bounds for them in terms of the classical characteristics from van Douwen's and Cichoń's diagrams and explain why in ZFC none of the characteristics from the diagrams is equal to nik or gr.

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