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 $\mu$ if and only if:

$(1): \quad$ $P$ 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.

Source of Name

This entry was named for Hans Hahn.

Sources

• 2013: Donald L. Cohn: Measure Theory (2nd ed.) ... (previous) ... (next): $4.1$