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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeMay 3rd 2024
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated: ## 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]]. <a href="https://ncatlab.org/nlab/revision/Hahn+decomposition+theorem/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Hahn+decomposition+theorem">current</a>

   Created:

   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.

   v1, current