Zdenek Silber: A countably tight P(K) space admitting a nonseparable measure
06/11/23 10:44
Tuesday, November 7, 2023 17:00
Location: room 601, Mathematical Institute, University of Wroclaw
Speaker: Zdenek Silber (IM PAN)
Title: A countably tight P(K) space admitting a nonseparable measure
Abstract: 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 \(L_1(\mu)\).
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 \([0,1]^{\omega_1}\). 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 \(P(K\times K)\) 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.
Location: room 601, Mathematical Institute, University of Wroclaw
Speaker: Zdenek Silber (IM PAN)
Title: A countably tight P(K) space admitting a nonseparable measure
Abstract: 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 \(L_1(\mu)\).
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 \([0,1]^{\omega_1}\). 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 \(P(K\times K)\) 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.