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.
Created on May 3, 2024 at 03:34:33. See the history of this page for a list of all contributions to it.