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I want to ask if it is true in general topological space that the countable union of sets of measure $0$ has $0$ measure?

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    It is true in general *measure spaces* as a consequence of countable additivity.2012-04-15
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    On the other hand, a countable union of *meager sets* is meager, in any topological space (meager means it is a countable union of nowhere dense subsets). Meager sets are the usual notion of "negligible" in the context of topological spaces, while "contained in a set of measure zero" is the usual notion of "negligible" in measure spaces.2012-04-15

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