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Do signed measures on sigma-rings always have a Hahn decomposition?

Let $X$ be a set. Let $\mathcal{R}$ be a set of subsets of $X$ such that $\{\} \in \mathcal{R}$ and For all members $A$ and $B$ of $\mathcal{R}$, $\;\; (A\cup B)-(A\cap B) \; \in \; \mathcal{R} ...