nLab Hahn decomposition theorem

Statement

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 $m\cap S=\emptyset$.

References

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