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Countable tightness in the spaces of regular probability measures

Volume 229 / 2015

Grzegorz Plebanek, Damian Sobota Fundamenta Mathematicae 229 (2015), 159-169 MSC: Primary 46E15, 46E27, 28A33; Secondary 54C35. DOI: 10.4064/fm229-2-4


We prove that if $K$ is a compact space and the space $P(K\times K)$ of regular probability measures on $K\times K$ has countable tightness in its weak$^*$ topology, then $L_1(\mu )$ is separable for every $\mu \in P(K)$. It has been known that such a result is a consequence of Martin's axiom MA$(\omega _1)$. Our theorem has several consequences; in particular, it generalizes a theorem due to Bourgain and Todorčević on measures on Rosenthal compacta.


  • Grzegorz PlebanekInstytut Matematyczny
    Uniwersytet Wrocławski
    Pl. Grunwaldzki 2/4
    50-384 Wrocław, Poland
  • Damian SobotaInstitute of Mathematics
    Polish Academy of Sciences
    Śniadeckich 8
    00-656 Warszawa, Poland

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