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



Support of Probability Measures on Separable Metric Spaces
A separable metric space is [strongly Lindelof](http://en.wikipedia.org/wiki/Lindelof_space), 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. 


