# 2020 seminar talk: Convergence of Borel measures and filters on omega

Talk held by Damian Sobota (KGRC) at the KGRC seminar on 2020-11-26.

### Abstract

The celebrated Josefson–Nissenzweig theorem asserts, under certain interpretations, that for every infinite compact space K there exists a sequence of normalized signed Borel measures on K which converges to 0 with respect to every continuous real-valued function (i.e. the corresponding integrals converge to 0). We showed that in the case of products of two infinite compact spaces K and L one can construct such a sequence of measures with an additional property that every measure has finite support—let us call such a sequence “an fsJN-sequence” (i.e. a finitely supported Josefson–Nissenzweig sequence). We then studied the case when the spaces K and L are only pseudocompact and we proved in ZFC that if the product of K and L is pseudocompact, then it also admits an fsJN-sequence. On the other hand, we showed that under the Continuum Hypothesis, or Martin's axiom, or even some weaker set-theoretic assumptions concerning weak P-points, there exists a pseudocompact space X such that its square is not pseudocompact and it does not admit any fsJN-sequences. During my talk I will discuss these as well as other results concerning the topic and obtained during a joint work with various combinations of J. Kakol, W. Marciszewski and L. Zdomskyy.

### Time and Place

Talk at 3:00pm via Zoom