The Hahn decomposition theorem

Abstract:

Let $(X,\mathcal {A},\mu )$ be a signed measure on the $\sigma$-algebra $\mathcal {A}$ 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 $E \subset N$.