Signed Measures and Radon-Nikodym - Examples and Constructions

The Radon-Nikodym theorem provides conditions under which one measure is the "density" of another, fundamental for probability theory and analysis.

TheoremRadon-Nikodym Theorem

Let $\mu$ be a $\sigma$ -finite measure and $\nu$ be a $\sigma$ -finite signed measure on $(X, \mathcal{F})$ . If $\nu \ll \mu$ (i.e., $\nu$ is absolutely continuous with respect to $\mu$ ), then there exists a $\mu$ -integrable function $f: X \to \mathbb{R}$ such that: $\nu(A) = \int_A f \, d\mu \text{ for all } A \in \mathcal{F}$

The function $f$ is unique up to sets of $\mu$ -measure zero. It is called the Radon-Nikodym derivative of $\nu$ with respect to $\mu$ , denoted: $f = \frac{d\nu}{d\mu}$

ExampleProbability Densities

In probability theory, if $X$ is a continuous random variable on $\mathbb{R}$ with distribution $\nu$ (a probability measure) that is absolutely continuous with respect to Lebesgue measure $\lambda$ , then by Radon-Nikodym: $\nu(A) = \int_A f(x) \, dx$

The function $f = \frac{d\nu}{d\lambda}$ is the probability density function (PDF) of $X$ .

For example, the normal distribution $\mathcal{N}(0, 1)$ has density: $f(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2}$

ExampleChange of Variables

If $\phi: \mathbb{R} \to \mathbb{R}$ is a diffeomorphism and $\lambda$ is Lebesgue measure, define $\nu(A) = \lambda(\phi^{-1}(A))$ . Then $\nu \ll \lambda$ and: $\frac{d\nu}{d\lambda}(x) = \left|\frac{d\phi^{-1}}{dx}(x)\right| = \frac{1}{|\phi'(\phi^{-1}(x))|}$

This is the classical change of variables formula from calculus, now understood through Radon-Nikodym.

Remark

The Lebesgue decomposition theorem complements Radon-Nikodym:

Lebesgue Decomposition Theorem: Let $\mu$ and $\nu$ be $\sigma$ -finite measures on $(X, \mathcal{F})$ . Then $\nu$ can be uniquely decomposed as: $\nu = \nu_{ac} + \nu_s$

where $\nu_{ac} \ll \mu$ (the absolutely continuous part) and $\nu_s \perp \mu$ (the singular part). By Radon-Nikodym, $\nu_{ac}(A) = \int_A f \, d\mu$ for some $f$ .

This decomposition separates a measure into its "smooth" part (with respect to $\mu$ ) and its "singular" part. For example, if $\nu = \lambda + \delta_0$ (Lebesgue measure plus a point mass), then:

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

The Radon-Nikodym theorem is foundational for defining conditional expectations in probability, densities in statistics, and understanding the structure of measures in analysis.