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Let $(X,M,\mu)$ be a complete measure space.

Does the set $\{\mu(E)|E\in M ,\mu(E)<\infty\}$ have to be a closed subset of $R$ ?

Thank you

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    @Randy: apply your proof to the counterexample of anonymous to see what's what.2012-12-09

2 Answers 2

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The answer is no. Take the positive integers (or any set containing them) and put a point mass of weight $1-\frac{1}{n}$ at every positive integer. Then $1$ is in the closure of your set but doesn't occur as the measure of any set.

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A partial answer is given by the fact that the range of a finite measure is closed. Also, the range is convex and hence, by the preceeding result, closed if the measure space is atomless.

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    Thank you. This is a useful partial answer2012-12-09