Fuzzy measures and discreteness (Q947802)

Let \((X,\mathcal{A})\) be a measurable space and let \(\mathcal F\) be the space of real valued set functions on \(\mathcal A\) of bounded variation, i.e., \(\|\mu\| =\sup \sum_{i=1}^n |\mu(A_i)-\mu(A_{i-1})|<\infty\), where the supremum is taken over all chains \(\emptyset = A_0 \subset A_1 \subset \cdots \subset A_n = X\), \(A_i \in \mathcal{A}\). It is proved that the space \((\mathcal{F},\|\cdot\|)\) is separable if and only if the \(\sigma\)-algebra \(\mathcal A\) is finite.