Definition:Hahn Decomposition

Definition
Let $\struct {X, \Sigma}$ be a measurable space.

Let $\mu$ be a signed measure on $\struct {X, \Sigma}$.

We say that $\tuple {P, N}$ is a Hahn decomposition of $\mu$ :


 * $(1): \quad$ $P$ and $N$ are disjoint.
 * $(2): \quad$ $X = P \cup N$
 * $(3): \quad$ $P$ is $\mu$-positive and $N$ is $\mu$-negative.

Also see

 * The Hahn Decomposition Theorem shows the existence of a Hahn decomposition of $\mu$, and a sense of "almost unique"-ness.