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
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
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.
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.