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Wydział Matematyki, Informatyki i Mechaniki Uniwersytetu Warszawskiego

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Sem. Topologia i T. Mnogości


A countably tight P(K) space admitting a nonseparable measure

Prelegent: Zdeněk Silber

2023-12-20 16:15

In the talk we focus on the relation of countable tightness of the space P(K) of Radon probabilty measures on a compact Hausdorff space K and of existence of measures in P(K) that have uncountable Maharam type. Recall that a topological space X has countable tightness if any element of the closure of a subset A of X lies in the closure of some countable subset of A. A Maharam type of a Radon probability measure mu is the density of the Banach space L1(mu).
It was proven by Fremlin that, under Martin's axiom and negation of continuum hypothesis, for a compact Hausdorff space K the existance of a Radon probability of uncountable type is equivalent to the exitence of a continuous surjection from K onto [0,1]^omega1. Hence, under such assumptions, countable tightness of P(K) implies that there is no Radon probability on K which has uncountable type. Later, Plebanek and Sobota showed that, without any additional set-theoretic assumptions, countable tightness of P(KxK) implies that there is no Radon probability on K which has uncountable type as well. It is thus natural to ask whether the implication "P(K) has countable tightness implies every Radon probability on K has countable type" holds in ZFC.
I will present our joint result with Piotr Koszmider that under diamond principle there is a compact Hausdorff space K such that P(K) has countable tightness but there exists a Radon probability on K of uncountable type.
Preprint with the result:

The next seminar meeting is scheduled for Wednesday, January 10.
Marry Christmas and Happy New Year!