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Let $(\Omega , \mathcal{A})$ be any measurable space, and let $(\mu_n)_n$ be a sequence of probability measures on this space, such that the limit $\mu(A) = \lim \mu_n(A)$ exists for any $A \in \mathcal{A}$.

Then $\mu$ is a finitely additive measure of total mass $1$ on $(\Omega , \mathcal{A})$. Is it always true that $\mu$ is $\sigma$-additive?

Asaf Karagila
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user69885
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    This is the Vitali-Hahn-Saks theorem: http://en.wikipedia.org/wiki/Vitali%E2%80%93Hahn%E2%80%93Saks_theorem –  Mar 28 '13 at 14:51
  • Thanks ! (and this is definitely an answer rather than a comment ) – user69885 Mar 28 '13 at 15:19
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    See also: http://math.stackexchange.com/questions/118592/limit-of-measures-is-again-a-measure – Martin Mar 28 '13 at 15:21

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