Hahn Decomposition Theorem

• categories: Measure Theory, Theorem

The Hahn Decomposition Theorem is a result in measure theory that applies to signed measures. It states that any measurable space can be divided into two disjoint sets: one on which a signed measure is non-negative and one on which it is non-positive.

Statement

Let $(X,\Sigma )$ be a measurable space, and let $\mu$ be a signed measure on $(X,\Sigma )$. Then, there exist two disjoint measurable sets $P,N\in \Sigma$ such that:

1. $P\cup N=X$ (the sets $P$ and $N$ form a partition of $X$),
2. $\mu (A)\ge 0$ for every measurable subset $A\subseteq P$,
3. $\mu (A)\le 0$ for every measurable subset $A\subseteq N$.

The sets $P$ and $N$ are called a Hahn decomposition of $X$ with respect to the signed measure $\mu$. This decomposition is not unique, but any two Hahn decompositions will differ only by a set of measure zero.

Key Implications

1. Jordan Decomposition
2. Total Variation

Outline of Proof

1. Supremum of Positive Sets: Define $P$ as the set where $\mu$ is “maximally positive” by considering subsets with non-negative measure and taking a supremum.
2. Complement as Negative Set: Define $N=X\setminus P$. By construction, $N$ will be the set where $\mu$ is “maximally negative.”
3. Verification: Show that $\mu$ is non-negative on all subsets of $P$ and non-positive on all subsets of $N$, completing the decomposition.

The Hahn Decomposition Theorem is foundational for further analysis in measure theory, particularly in the study of signed and complex measures.