What can the range of a measure be?

If it's a finite set, there must be $a_1, \ldots, a_m > 0$ such that $\mu[\mathcal A] = \{ \sum_i a_i x_i : x \in \{0,1\}^m\}$.


It is a well know result by Saks (also generalized by Lyapunov) that if $\mu$ has no atoms, then $\mu$ can take any value in $[0,\mu[X])$. If $\mu$ has atoms, the range of $\mu$ can get a little weird.

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Measure Theory

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