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If a measure $P$ can take value of $+\infty $, does countable additivity ($\sigma$-additivity) imply continuity at $\emptyset$? If not, what is a good example?

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Not set-theory. Plus, $P$ is not a probability, so the tag should be measure-theory instead.) – Andrés Caicedo Jan 23 '13 at 22:56
The reals with Lebesgue measure form a counterexample, right? Take $A_n=(n,\infty)$. Their intersection is empty. – Andrés Caicedo Jan 23 '13 at 22:57

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