= Hahn decomposition
{c}
{title2=$X=P\sqcup N$}
A <Hahn decomposition> of a <signed measure> partitions its space into <measurable sets> $P,N$ such that every measurable subset of $P$ has nonnegative <measure> and every measurable subset of $N$ has nonpositive <measure>. For finite <total variation measure>, set $\mu^+=(|\mu|+\mu)/2$ and choose a $\mu^+$-<null set> $N$ maximizing $|\mu|(N)$: a <countable union> of approximating <null sets> attains the <supremum>. If some subset of its complement has negative signed mass, successively remove disjoint subsets with positive mass at least half the <supremum> still available. Their masses tend to zero because their sum is bounded by total variation. The remainder has no positive-<measure> subset, has negative mass, and is $\mu^+$-null, contradicting maximality after adjoining it to $N$. On $P$ one has $|\mu|=\mu$ and on $N$ one has $|\mu|=-\mu$. Consequently $d\mu^+/d|\mu|=\mathbf1_P$, an explicit <Radon-Nikodym derivative>.
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