Kurt Gödel Research Center

2015 seminar talk: The Nikodym property and cardinal invariants of the continuum

Talk held by Damian Sobota (Polish Academy of Sciences, Warsaw) at the KGRC seminar on 2015-12-10.

Abstract

A Boolean algebra $\mathcal{A}$ is said to have the Nikodym property if every sequence $(\mu_n)$ of measures on $\mathcal{A}$ which is elementwise bounded (i.e. $\sup_n|\mu_n(a)|<\infty$ for every $a\in\mathcal{A}$) is uniformly bounded (i.e. $\sup_n\|\mu_n\|<\infty$). The property is closely related to the classical Banach-Steinhaus theorem for Banach spaces.

My recent study concerns the problem how (and whether at all) we can describe the structure of the class of Boolean algebras with the Nikodym property in terms of well-known objects occuring inside $\wp(\omega)$ or $\omega^\omega$, e.g. countable Boolean algebras, dominating families, Lebesgue null sets etc. During my talk I will present an attempt to obtain such a description via families of antichains in countable subalgebras of $\wp(\omega)$ having some special measure-theoretic properties.

If time allows, I will present some consequences of my research for the Efimov problem and C*-algebra theory.