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# Shorter Notes: The Hahn Decomposition Theorem

Raouf Doss
Proceedings of the American Mathematical Society
Vol. 80, No. 2 (Oct., 1980), p. 377
DOI: 10.2307/2042981
Stable URL: http://www.jstor.org/stable/2042981
Page Count: 1
Let $(X, \mathscr{A}, \mu)$ be a signed measure on the $\sigma$-algebra $\mathscr{A}$ of subsets of $X$. We give a very short proof of the Hahn decomposition theorem, namely, that $X$ can be partitioned into two subsets $P$ and $N$ such that $P$ is positive: $\mu(E) \geqslant 0$ for every $E \subset P$, and $N$ is negative: $\mu(E) \leqslant 0$ for every $E \subset N$.