# Support of Probability Measures on Separable Metric Spaces

Let $X$ be a separable metric space and $p$ a probability measure on the Borel Sets of $X$.

Denote $S_p$ the support of $p$, i.e. the set of points which have positive measure for any ball around them

How to prove that the support of p is of full measure, i.e. $p(S_p)=1$?

Thanks, Shlomi

-
This could also be of interest: mathoverflow.net/questions/44408/… –  Byron Schmuland Apr 23 '11 at 15:49

## 1 Answer

A separable metric space is strongly Lindelof, that is, every open cover of an open subset has a countable subcover. The complement of the support is the union of all open balls with zero measure. By reducing to a countable subcover, we see that this set has measure zero. So the support has full measure.

-