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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.

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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.