During my lecture I would like to focus on the following problem from the boundary of measure theory and set-theoretical topology. Let K be a compact Hausdorff space. Assume that the space P(K) of all regular probability measures has countable tightness. Does this imply that every measure mu in P(K) is of countable Maharam type (ie. L_1(mu) is separable)? Fremlin answered this in affirmative assuming Martin's Axiom and Talagrand proved in ZFC that if the space P(K) has tightness at most omega_1, then every measure is of type at most omega_1 as well. Together with Grzegorz Plebanek we have obtained in ZFC the following result: if P(KxK) has countable tightness, then every mu in P(K) has countable type. This does not answer the question entirely, but seems to be an important step forward. After the proof of the theorem I would like to present some consequences concerning eg. the property (C) of Corson or measures on Rosenthal compacta.
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