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Davide Giraudo
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Topological nessary and sufficient condition for tightness

Recall the definition of tightness for a probability measure $\mathbb P$ on the Borel $\sigma$-algebra of a metric space $(S,d)$:

For each $\varepsilon>0$, we can find a compact subset $K$ of $X$ such that $\mathbb P(K)\geq 1-\varepsilon$.

The question is: is there a "nice" topological characterization of metric spaces such that each Borel probability measure is tight?

In Billingsley's book Convergence of probability measures, 1968, it's said that it's an open problem. I wish know whether some progress have been done so far.

Call EPT a metric space on which each Borel probability measure is tight. Some remarks:

  • By Ulam's theorem, each separable metric space topologically complete is EPT.
  • A necessary and sufficient condition that each probability has a separable support is that each subset of $D$ of $S$ which is discrete have a non-measurable cardinal (i.e. we can't find a probability measure $\mu$ on $2^D$ such that $\mu(\{x\})=0$ for each $x\in D$). Hence for each EPT space, each discrete subset have a non-measurable cardinal.
Davide Giraudo
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