# 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

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