Piotr Borodulin-Nadzieja — Page 2

arXiv2012

Sequential closure in the space of measures

We show that there is a compact topological space carrying a measure which is not a weak* limit of finitely supported measures but is in the sequential closure of the set of such measures. We construct compact spaces with measures of arbitrarily high levels of complexity in this sequential hierarchy. It follows that there is a compact space in which the sequential closure cannot be obtained in countably many steps. However, we show that this is not the case for our spaces where the sequential closure is always obtained in countably many steps.

math.GN↗

arXiv2011

On the isomorphism problem for measures on Boolean algebras

The paper investigates possible generalisations of Maharam's theorem to a classification of Boolean algebras that support a finitely additive measure. We prove that Boolean algebras that support a finitely additive non-atomic uniformly regular measure are metrically isomorphic to a subalgebras of the Jordan algebra with the Lebesgue measure. We give some partial analogues to be used for a classification of algebras that support a finitely additive non-atomic measure with a higher uniform regularity number. We show that some naturally induced equivalence relations connected to metric isomorphism are quite complex even on the Cantor algebra and therefore probably we cannot hope for a nice general classification theorem for finitely--additive measures. We present an example of a Boolean algebra which supports only separable measures but no uniformly regular one.

math.LO↗