A measure which is not tight

Asked Oct 26 '23 at 15:10

Active Oct 26 '23 at 15:10

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I know that if $(E,d)$ is a metric space complete and separable (metric polish space), then every single probability $\mu$ form a tight family, meaning for all $\epsilon>0$ there is a compact $K$ such that $\mu(K^c)\leq \epsilon.$

I would like to see counterexample of a single probability which is not tight, if $E$ is non-complete.

For instance, I was thinking of taking $E = \mathbb{Q} \cap ]0,1[$, or the dyadic rational numbers. It is separable but non-complete. Alternatively, I was trying to think of a non-complete metric space which has "trivial" compacts, but my lack of topology does not help me.

asked Oct 26 '23 at 15:10

Xefeer

1

To be clear: you ask for a separable metric space (not complete) for this example. So we cannot use the "lower limit topology" as here https://en.m.wikipedia.org/wiki/Inner_regular_measure – GEdgar Oct 26 '23 at 15:58
See also the last bullet point in this question – spaceisdarkgreen Oct 26 '23 at 16:46
Yes thank you @GEdgar. Great example. – Xefeer Oct 26 '23 at 20:42

A measure which is not tight

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