Signed Measures and Radon-Nikodym - Core Definitions

Signed measures generalize ordinary measures by allowing negative values, providing a framework for understanding differences of measures and integration with respect to such differences.

DefinitionSigned Measure

A signed measure on a measurable space $(X, \mathcal{F})$ is a function $\nu: \mathcal{F} \to [-\infty, \infty]$ such that:

$\nu(\emptyset) = 0$
$\nu$ takes at most one of the values $\pm\infty$
For any countable collection $\{A_i\}$ of pairwise disjoint sets in $\mathcal{F}$ : $\nu\left(\bigcup_{i=1}^{\infty} A_i\right) = \sum_{i=1}^{\infty} \nu(A_i)$

where the series converges absolutely if $\nu$ is finite.

The condition that $\nu$ cannot take both $+\infty$ and $-\infty$ ensures that sums like $\infty - \infty$ never arise.

ExampleBasic Signed Measures

Difference of measures: If $\mu$ and $\nu$ are finite measures on $(X, \mathcal{F})$ , then $\mu - \nu$ is a signed measure.
Integration: For an integrable function $f \in L^1(\mu)$ , define: $\nu(A) = \int_A f \, d\mu$

Then $\nu$ is a signed measure. If $f$ changes sign, so does $\nu$ .

Charge distribution: In physics, a charge distribution can be positive (positive charges) or negative (negative charges), naturally giving rise to a signed measure.

DefinitionAbsolute Continuity

Let $\mu$ and $\nu$ be signed measures on $(X, \mathcal{F})$ . We say $\nu$ is absolutely continuous with respect to $\mu$ , written $\nu \ll \mu$ , if: $\mu(A) = 0 \Rightarrow \nu(A) = 0$ for all $A \in \mathcal{F}$ .

Intuitively, $\nu$ is absolutely continuous with respect to $\mu$ if $\nu$ assigns zero measure to any set that $\mu$ considers negligible.

DefinitionMutually Singular Measures

Two signed measures $\mu$ and $\nu$ are mutually singular, written $\mu \perp \nu$ , if there exist disjoint sets $A, B \in \mathcal{F}$ with $A \cup B = X$ such that:

$|\mu|(B) = 0$ (i.e., $\mu$ is concentrated on $A$ )
$|\nu|(A) = 0$ (i.e., $\nu$ is concentrated on $B$ )

Here $|\mu|$ denotes the total variation of $\mu$ (defined below).

Remark

Key observations:

Every ordinary measure is a signed measure: Non-negative measures are special cases where $\nu$ never takes negative values.
Signed measures form a vector space: If $\nu_1$ and $\nu_2$ are signed measures and $\alpha, \beta \in \mathbb{R}$ , then $\alpha \nu_1 + \beta \nu_2$ is a signed measure (with appropriate finiteness conditions).
Integration with respect to signed measures: One can integrate with respect to $\nu$ by decomposing it into positive and negative parts.