206 research outputs found
A dichotomy for the convex spaces of probability measures
We show that every nonempty compact and convex space M of probability Radon
measures either contains a measure which has `small' local character in M or
else M contains a measure of `large' Maharam type. Such a dichotomy is related
to several results on Radon measures on compact spaces and to some properties
of Banach spaces of continuous functions.Comment: 10 page
Countable tightness in the spaces of regular probability measures
We prove that if is a compact space and the space of
regular probability measures on has countable tightness in its
topology, then is separable for every . It has
been known that such a result is a consequence of Martin's axiom
MA.
Our theorem has several consequences; in particular, it generalizes a theorem
due to Bourgain and Todor\v{c}evi\'c on measures on Rosenthal compacta.Comment: 9 page
On Efimov spaces and Radon measures
We give a construction under CH of an infinite Hausdorff compact space having no converging sequences and carrying no Radon measure of uncountable type. Under ? we obtain another example of a compact space with no convergent sequences, which in addition has the stronger property that every nonatomic Radon measure on it is uniformly regular. This example refutes a conjecture of Mercourakis from 1996 stating that if every measure on a compact space K is uniformly regular then K is necessarily sequentially compact
On measures on Rosenthal compacta
We show that if K is Rosenthal compact which can be represented by functions
with countably many discontinuities then every Radon measure on K is countably
determined. We also present an alternative proof of the result stating that
every Radon measure on an arbitrary Rosenthal compactum is of countable type.
Our approach is based on some caliber-type properties of measures,
parameterized by separable metrizable spaces.Comment: 14 page
Precalibre pairs of measure algebras
AbstractWe consider Radon measures μ and pairs (κ,λ) of cardinals such that among every κ many positive measure sets there are λ many whose intersection is nonempty. Such families are connected with the cardinal invariants of the ideal of μ-null sets and have found applications in various subjects of topological measure theory. We survey many of such connections and applications and give some new ones. In particular we show that it is consistent to have a Corson compact space carrying a Radon measure of type c>ℵ1 and we partially answer a question of Haydon about measure precalibres
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