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If $X$ is a set, $M$ is a σ-algebra on $X$, and $\mu$ is a signed measure, i.e., a countably additive functional $M\to\mathbf{R}$, then $\mu$ is bounded and there is $S\in M$ such that

$\mu(m)\ge0$ for every $m\in M$ such that $m\subset S$;

$\mu(m)\le0$ for every $m\in M$ such that $m\cap S=\emptyset$.

A standard theorem present in many introductory textbooks. See, for example, Theorem 231E in Fremlin’s Measure Theory.

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