Signed Measures and Radon-Nikodym - Applications

TheoremConditional Expectation via Radon-Nikodym

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and $X$ be an integrable random variable ( $X \in L^1(\mathbb{P})$ ). Let $\mathcal{G} \subseteq \mathcal{F}$ be a sub-sigma-algebra.

Define a signed measure $\nu$ on $(\Omega, \mathcal{G})$ by: $\nu(A) = \int_A X \, d\mathbb{P} \text{ for } A \in \mathcal{G}$

Then $\nu \ll \mathbb{P}|_{\mathcal{G}}$ , and by Radon-Nikodym, there exists a $\mathcal{G}$ -measurable function $Y$ such that: $\nu(A) = \int_A Y \, d\mathbb{P} \text{ for all } A \in \mathcal{G}$

This function $Y$ is the conditional expectation of $X$ given $\mathcal{G}$ , denoted $\mathbb{E}[X | \mathcal{G}]$ .

This theorem shows that conditional expectation is essentially the Radon-Nikodym derivative, providing a rigorous foundation for conditional probability and expectation in probability theory.

ExampleComputing Conditional Expectations

Let $X$ and $Y$ be jointly distributed random variables on $\mathbb{R}^2$ with joint density $f_{X,Y}(x, y)$ .

The conditional expectation $\mathbb{E}[Y | X = x]$ can be computed via Radon-Nikodym. Define: $\nu(A) = \int_A y f_{X,Y}(x, y) \, dy$

The marginal density of $X$ is $f_X(x) = \int f_{X,Y}(x, y) \, dy$ . By Radon-Nikodym: $\mathbb{E}[Y | X = x] = \frac{d\nu}{d f_X}(x) = \frac{\int y f_{X,Y}(x, y) \, dy}{f_X(x)}$

This is the familiar formula for conditional expectation in terms of densities.

Remark

Applications of Radon-Nikodym in various fields:

Likelihood ratios: In statistics, the likelihood ratio between two probability measures $\mathbb{P}$ and $\mathbb{Q}$ (with $\mathbb{Q} \ll \mathbb{P}$ ) is: $L = \frac{d\mathbb{Q}}{d\mathbb{P}}$

This is fundamental for hypothesis testing and Bayesian inference.

Girsanov's theorem: In stochastic calculus, changing probability measures (to eliminate drift in SDEs) uses Radon-Nikodym derivatives: $\frac{d\mathbb{Q}}{d\mathbb{P}} = \exp\left(\int_0^T \theta_s \, dW_s - \frac{1}{2}\int_0^T \theta_s^2 \, ds\right)$
Martingale theory: Martingales can be characterized via Radon-Nikodym derivatives of conditional distributions.
Information theory: The Kullback-Leibler divergence between measures involves the Radon-Nikodym derivative: $D_{KL}(\mathbb{Q} || \mathbb{P}) = \int \log\left(\frac{d\mathbb{Q}}{d\mathbb{P}}\right) d\mathbb{Q}$
Optimal stopping: The value function in optimal stopping problems can be expressed using Radon-Nikodym derivatives of stopped processes.

The Radon-Nikodym theorem's applications span probability, statistics, finance, and physics, making it one of the most practically important results in measure theory.