Signed Measures and Radon-Nikodym - Key Properties

Signed measures admit a unique decomposition into positive and negative parts, enabling integration and analysis through their constituent measures.

TheoremHahn Decomposition Theorem

Let $\nu$ be a signed measure on $(X, \mathcal{F})$ . There exist disjoint sets $P, N \in \mathcal{F}$ with $P \cup N = X$ such that:

For every $A \in \mathcal{F}$ with $A \subseteq P$ : $\nu(A) \geq 0$ (P is a positive set)
For every $A \in \mathcal{F}$ with $A \subseteq N$ : $\nu(A) \leq 0$ (N is a negative set)

The pair $(P, N)$ is called a Hahn decomposition of $X$ with respect to $\nu$ . It is not unique, but any two Hahn decompositions differ by a set of $|\nu|$ -measure zero.

The Hahn Decomposition shows that any signed measure naturally splits the space into regions where it is positive and negative. This geometric insight is powerful for understanding the structure of signed measures.

DefinitionJordan Decomposition

Given a signed measure $\nu$ and a Hahn decomposition $(P, N)$ , define: $\nu^+(A) = \nu(A \cap P), \quad \nu^-(A) = -\nu(A \cap N)$

Then $\nu^+$ and $\nu^-$ are non-negative measures called the positive and negative variations of $\nu$ . We have: $\nu = \nu^+ - \nu^-$

The measure $|\nu| = \nu^+ + \nu^-$ is called the total variation of $\nu$ .

This decomposition is called the Jordan decomposition of $\nu$ .

ExampleComputing Jordan Decomposition

Let $\nu(A) = \int_A f \, d\mu$ where $f \in L^1(\mu)$ . Then:

$\nu^+(A) = \int_A f^+ \, d\mu$ where $f^+ = \max(f, 0)$
$\nu^-(A) = \int_A f^- \, d\mu$ where $f^- = \max(-f, 0)$
$|\nu|(A) = \int_A |f| \, d\mu$

For instance, if $f(x) = \cos(x)$ on $[0, 2\pi]$ with Lebesgue measure:

$P = [0, \pi/2] \cup [3\pi/2, 2\pi]$ (where $\cos(x) \geq 0$ )
$N = [\pi/2, 3\pi/2]$ (where $\cos(x) \leq 0$ )
$\nu^+([0, 2\pi]) = \int_P \cos(x) \, dx = 2$
$\nu^-([0, 2\pi]) = \int_N |\cos(x)| \, dx = 2$
$|\nu|([0, 2\pi]) = 4$

Remark

Important properties of the Jordan decomposition:

Uniqueness: The measures $\nu^+$ and $\nu^-$ are uniquely determined by $\nu$ , even though the Hahn decomposition $(P, N)$ is not unique.
Mutual singularity: $\nu^+ \perp \nu^-$ (they are concentrated on disjoint sets).
Minimality: $|\nu|$ is the smallest positive measure $\mu$ such that $|\nu(A)| \leq \mu(A)$ for all $A \in \mathcal{F}$ .
Integration: For $f$ integrable with respect to $|\nu|$ : $\int f \, d\nu = \int f \, d\nu^+ - \int f \, d\nu^-$