# Topological necessary 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.
• If we assume the metric space separable, we have the answer from Dudley's book: each probability measure on $S$ is tight if and only if $S$ is universally measurable (that is, if $\widehat S$ is the metric completion of $S$, then $S$ is $\mathbb P$-measurable for each probability measure $\mathbb P$ on $\widehat S$).
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Done now: mathoverflow.net/questions/99497/…. – Davide Giraudo Jun 13 '12 at 20:19
The question whether there is a discrete EPT is equivalent to the question of the existence of a non-measurable cardinal. Deciding this seems to be beyond the scope of ZFC. You might be interested in having a look at Section 438 in Vol 4I of Fremlin's measure theory. In 438H Fremlin proves (in your terminology) that a complete metric space $X$ is EPT if and only if the minimal cardinality of a basis of the topology is a non-measurable cardinal. Also, have a look at the numerous passages on the Banach-Ulam problem in these books. – t.b. Jun 15 '12 at 7:03
A remark on terminology: Fremlin calls a Hausdorff space in which every finite Borel measure is tight a "Radon space" which seems to have been investigated quite deeply. See also chapter 434 of the book. (Unfortunately, there are about as many notions of Radon measures as there are authors (probably more...), so "Radon space" has various meanings, compare e.g. here and here.) – t.b. Jun 15 '12 at 7:06
@t.b. Thank you very much for the references. I will look them carefully. – Davide Giraudo Jun 15 '12 at 8:33