Signed Measures and Radon-Nikodym - Main Theorem

TheoremLebesgue Decomposition Theorem

Let $\mu$ and $\nu$ be $\sigma$ -finite measures on a measurable space $(X, \mathcal{F})$ . Then there exists a unique decomposition: $\nu = \nu_{ac} + \nu_s$

where:

$\nu_{ac} \ll \mu$ (absolutely continuous with respect to $\mu$ )
$\nu_s \perp \mu$ (mutually singular with $\mu$ )

By the Radon-Nikodym theorem, there exists $f \geq 0$ such that: $\nu_{ac}(A) = \int_A f \, d\mu \text{ for all } A \in \mathcal{F}$

The decomposition $\nu = \nu_{ac} + \nu_s$ is unique in the sense that if $\nu = \nu_1 + \nu_2$ with $\nu_1 \ll \mu$ and $\nu_2 \perp \mu$ , then $\nu_1 = \nu_{ac}$ and $\nu_2 = \nu_s$ .

The Lebesgue Decomposition separates any measure into components that are "compatible" with $\mu$ (the absolutely continuous part) and "incompatible" with $\mu$ (the singular part). This is fundamental for understanding the structure of measures.

ExampleDiscrete and Continuous Parts

On $\mathbb{R}$ with Lebesgue measure $\lambda$ , consider: $\nu = \frac{1}{2}\lambda + \frac{1}{2}\delta_0$

where $\delta_0$ is the Dirac measure at $0$ . Then:

$\nu_{ac} = \frac{1}{2}\lambda$ (absolutely continuous with respect to $\lambda$ )
$\nu_s = \frac{1}{2}\delta_0$ (singular with respect to $\lambda$ )

The Radon-Nikodym derivative is: $\frac{d\nu_{ac}}{d\lambda} = \frac{1}{2}$

This models a probability distribution with both continuous and discrete components.

Remark

Important consequences and applications:

Every measure decomposes: Any measure $\nu$ on $\mathbb{R}$ can be written as $\nu = \nu_{ac} + \nu_{sc} + \nu_d$ where:
- $\nu_{ac}$ is absolutely continuous (has a density)
- $\nu_{sc}$ is singular continuous (like the Cantor-Lebesgue function)
- $\nu_d$ is discrete (sum of point masses)
Probability distributions: In probability, every distribution function $F$ decomposes into absolutely continuous (with PDF), singular continuous, and jump components.
Spectral theory: The spectral measure of a self-adjoint operator decomposes into absolutely continuous, singular continuous, and pure point spectra.
Harmonic analysis: The Lebesgue decomposition helps classify measures in terms of their Fourier transform properties.
Optimal transport: The decomposition is used to understand the structure of optimal transport plans.

The Lebesgue Decomposition Theorem, together with Radon-Nikodym, provides a complete understanding of how two measures relate to each other, forming a cornerstone of modern measure theory.