Small Boolean algebras with the Nikodym property

Speaker(s)

Damian Sobota

Affiliation

doktorant IMPAN

Date

May 27, 2015, 4:15 p.m.

Room

room 5050

Seminar

Topology and Set Theory Seminar

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Let A be a Boolean algebra. A sequence of measures (mu_n) on A is: * pointwise bounded if the sequence (|mu_n(a)|) is bounded for every a in A; * uniformly bounded if the sequence (||mu_n||) is bounded (|| || denotes the variation of a measure). We say that A has the Nikodym property if every pointwise bounded sequence of measures on A is uniformly bounded. A standard examples of algebras with the Nikodym property are sigma-complete algebras (the Nikodym theorem) or the algebra of Jordan-measurable subsets of [0,1] (Schachermayer's theorem). All known so far examples of such algebras have been of cardinality at least the continuum c. We thus ask: is there a consistent example of a Boolean algebra with the Nikodym property and cardinality less than c? The plan of the talk is as follows. First, I will show that an algebra with the Nikodym property must be of size at least max(b,s,cov(M)), where b is the bounding number, s is the splitting number and cov(M) denotes the covering of category. Second, I will answer the question in affirmative -- assuming that the cofinality of measure cof(N) is kappa