Signed measure

A signed measure is a countably additive set function on a sigma-algebra that permits negative values. In the extended-real convention it takes values in $[-\infty ,\infty ]$, has value zero on the empty set, and does not take both $+\infty$ and $-\infty$, so its additive sums are defined. The important finite case is real-valued on every measurable set.^[ref]

For distinct points $a,b$, the difference of Dirac measures $\nu ={\delta }_a-2{\delta }_b$ is a finite signed measure. It has $\nu (\{a\})=1$, $\nu (\{b\})=-2$ and $\nu (\{a,b\})=-1$. Its variation measure is $|\nu |={\delta }_a+2{\delta }_b$, whose total mass is $3$, rather than the absolute value $1$ of its signed total mass. The total variation norm of a measure records both contributions before cancellation.

A Hahn decomposition separates a signed measure into a region on which all measurable subsets have nonnegative mass and a region on which all measurable subsets have nonpositive mass. It is the setwise counterpart of splitting an integrable function into its positive part and negative part.

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 $d{\mu }^{+}/d|\mu |=1_P$, an explicit Radon-Nikodym derivative.