1 Statement of the Radon–Nikodym Theorem

1.1 Absolute continuity and its meaning

Let \((X,\Sigma)\) be a measurable space and let \(\mu,\nu\) be \(\sigma\)-finite measures on it. The relation \(\mu \ll \nu\) (read “\(\mu\) is absolutely continuous with respect to \(\nu\)”) means that whenever \(\nu(A)=0\) for a measurable set \(A\in\Sigma\), then also \(\mu(A)=0\). Intuitively, \(\nu\) does not ignore any measurable sets that \(\mu\) regards as having positive mass.

1.2 Existence of the derivative

The Radon–Nikodym theorem states that if \(\mu \ll \nu\) and \(\nu\) is \(\sigma\)-finite, then there exists a \(\nu\)-measurable function \(f\) such that for every measurable set \(A\in\Sigma\), \[ \mu(A)=\int_A f\,d\nu. \] Such a function \(f\) is called a Radon–Nikodym derivative of \(\mu\) with respect to \(\nu\).

1.3 Uniqueness up to ν-almost everywhere equality

The derivative is not unique pointwise. If \(f\) and \(g\) both satisfy \(\mu(A)=\int_A f\,d\nu=\int_A g\,d\nu\) for all \(A\), then \(f=g\) holds \(\nu\)-almost everywhere. This “almost everywhere” ambiguity reflects the fact that \(\nu\)-null sets do not affect integrals with respect to \(\nu\).

1.4 Integrated form of the derivative

The theorem also yields an integrated characterization for general measurable functions. Whenever \(f=\frac{d\mu}{d\nu}\) and \(h\) is measurable with \(\inth\,d\mu<\infty\), one can express

\[ \int_X h\,d\mu=\int_X h\,\frac{d\mu}{d\nu}\,d\nu. \] This is the practical computational form used in applications.

2 Definition and Notation

2.1 Radon–Nikodym derivative as dμ/dν

Given \(\mu\ll \nu\), the Radon–Nikodym derivative is denoted by \[ \frac{d\mu}{d\nu}, \] and refers to any representative of the \(\nu\)-a.e. equivalence class of functions \(f\) satisfying \(\mu(A)=\int_A f\,d\nu\). The notation is suggestive of a “density,” though \(\mu\) and \(\nu\) need not be related to a common base density in the classical sense.

2.2 Measurability and representative functions

The derivative \(f\) is required to be \(\nu\)-measurable. Different representatives may differ on sets where \(\nu\) is zero; these changes do not alter any integral \(\int_A f\,d\nu\). As a result, many statements are formulated “up to \(\nu\)-almost everywhere equality.”

2.3 Relationship to densities in special cases

2.3.1 Derivatives for measures with densities

If \(\nu\) is a reference measure and \(\mu\) is given by a density \(g\) with respect to \(\nu\), meaning \(\mu(A)=\int_A g\,d\nu\), then the Radon–Nikodym derivative is exactly \(g\) (up to \(\nu\)-a.e. equality). This recovers the familiar calculus concept of a derivative of measures.

2.3.2 Derivatives on discrete spaces

On a countable set \(X\) with the power set \(\Sigma\), measures correspond to sequences. If \(\nu(\{x\})>0\) wherever \(\mu(\{x\})>0\), then \[ \frac{d\mu}{d\nu}(x)=\frac{\mu(\{x\})}{\nu(\{x\})} \] on points where \(\nu(\{x\})>0\). The Radon–Nikodym formula then reduces to a weighted summation.

3 Connection to Likelihood Ratios and Changes of Measure

3.1 Likelihood ratio interpretation

In probability, when \(\mu\) and \(\nu\) are probability measures and \(\mu\ll \nu\), the Radon–Nikodym derivative acts like a likelihood ratio. For an event \(A\), \[ \mu(A)=\int_A \frac{d\mu}{d\nu}\,d\nu, \] so \(\frac{d\mu}{d\nu}\) quantifies how much more (or less) likely sets are under \(\mu\) compared with \(\nu\), relative to the reference measure.

3.2 Using the derivative to compute expectations

For an integrable random variable \(H\) (measurable function on the underlying space), \[ \mathbb{E}_\mu[H]=\int H\,d\mu=\int H\,\frac{d\mu}{d\nu}\,d\nu=\mathbb{E}_\nu\!\left[H\,\frac{d\mu}{d\nu}\right]. \] This identity allows one to compute expectations under \(\mu\) by integrating under \(\nu\) with a correction factor.

3.3 Change-of-measure formula

3.3.1 Transforming integrals under ν

A standard form of the change-of-measure rule is: \[ \int_X \varphi\,d\mu=\int_X \varphi\,\frac{d\mu}{d\nu}\,d\nu \] for measurable \(\varphi\) for which at least one side is well-defined. The factor \(\frac{d\mu}{d\nu}\) is the bridge between the two measures.

3.3.2 Applications in probability modeling

In modeling, one often starts with an analytically convenient measure \(\nu\) and introduces a target distribution \(\mu\). The Radon–Nikodym derivative enables rewriting quantities computed under the target distribution in terms of computations under the reference distribution, which can simplify analysis and simulation.

4 Radon–Nikodym Derivative and Conditional Expectation

4.1 Conditional expectation as a “density” concept

Conditional expectation can be viewed through a measure-theoretic lens that resembles differentiation. While the conditional expectation \( \mathbb{E}_\mu[\cdot \mid \mathcal{G}] \) is not literally a Radon–Nikodym derivative in general, it plays an analogous role by specifying how averaging under \(\mu\) changes when restricted to a sub-\(\sigma\)-algebra \(\mathcal{G}\).

4.2 The measure induced by conditional distributions

Given a probability space \((X,\Sigma,\mu)\) and a sub-\(\sigma\)-algebra \(\mathcal{G}\), conditioning can be represented by constructing, for each outcome, a probability measure describing the conditional law of variables given \(\mathcal{G}\). These conditional measures are often related to Radon–Nikodym derivatives when densities exist relative to appropriate reference measures.

4.2.1 Regular conditional probabilities (overview level)

Under standard hypotheses (e.g., when the space is sufficiently nice, such as being a standard Borel space), one can define regular conditional probabilities: for each \( \omega \), a measure \(P_\omega\) such that \(P_\omega(A)\) behaves measurably in \(\omega\) and satisfies the usual conditioning identities. When \(P_\omega\) is absolutely continuous with respect to a fixed reference measure, Radon–Nikodym derivatives provide density functions for these conditional laws.

In stochastic processes, Radon–Nikodym derivatives frequently appear as density processes that convert one probability measure into another over time. Martingale properties ensure that the evolving likelihood ratios behave consistently with the filtration, enabling rigorous formulation of measure changes for processes.

5 Key Examples

5.1 Lebesgue measure vs. absolutely continuous measures

On \(\mathbb{R}^n\), let \(\nu\) be Lebesgue measure. If \(\mu\) is absolutely continuous with respect to Lebesgue measure, then there exists an integrable function \(f\) such that \(\mu(A)=\int_A f\,dx\). In this case, \(\frac{d\mu}{d\nu}=f\), recovering the classical notion of a density.

5.2 Derivatives under pushforward mappings

5.2.1 Computing derivatives via transformations

Let \(T:X\to Y\) be measurable, and consider the pushforward measures \(\nu\circ T^{-1}\) and \(\mu\circ T^{-1}\) on \(Y\). When absolute continuity holds between the pushforwards, the Radon–Nikodym derivative on \(Y\) can sometimes be computed from Jacobian-type factors or from how \(T\) distorts mass. Exact formulas depend on structure (e.g., differentiability and nondegeneracy), but the guiding idea is that the derivative transfers through the transformation while accounting for how sets are mapped.

5.3 Singular vs. absolutely continuous components (decomposition)

Not every measure \(\mu\) is absolutely continuous with respect to a given \(\nu\). The Lebesgue decomposition theorem (discussed elsewhere in this entry) states that \(\mu\) can be separated into an absolutely continuous part and a singular part relative to \(\nu\). The Radon–Nikodym derivative typically represents only the absolutely continuous component, while the singular component has no density with respect to \(\nu\).

6.1 Hahn–Jordan decomposition (contrast with RN)

The Hahn–Jordan decomposition splits a signed measure into the difference of two mutually singular nonnegative measures. By contrast, the Radon–Nikodym theorem provides a density for one measure relative to another under absolute continuity. Conceptually, both results decompose or represent measures, but they address different structures: signed measures versus measure domination relationships.

6.2 Lebesgue decomposition theorem overview

The Lebesgue decomposition theorem generalizes the Radon–Nikodym perspective by decomposing \(\mu\) into \(\mu=\mu_{\mathrm{ac}}+\mu_{\mathrm{s}}\), where \(\mu_{\mathrm{ac}}\ll \nu\) and \(\mu_{\mathrm{s}}\perp \nu\). The Radon–Nikodym derivative applies to \(\mu_{\mathrm{ac}}\), producing the density for the part of \(\mu\) that is absolutely continuous relative to \(\nu\).

6.3 Comparison with mutual singularity

If \(\mu\) and \(\nu\) are mutually singular, then each concentrates on disjoint measurable sets in the sense of null sets. In that situation, \(\mu\not\ll \nu\), so a Radon–Nikodym derivative with respect to \(\nu\) does not exist. This contrast clarifies the central role of absolute continuity in enabling differentiation of measures.

6.4 Bayes’ rule in measure-theoretic terms (interpretive)

Bayes’ rule can be expressed using Radon–Nikodym derivatives by treating prior and posterior measures as probability measures on a sample space or parameter space. When relevant conditional distributions are absolutely continuous with respect to suitable reference measures, the ratio of densities corresponds to Bayes’ updating through likelihood and normalization factors.

7 Computation and Practical Considerations

7.1 Finding dμ/dν in common settings

In many applications, \(\frac{d\mu}{d\nu}\) is obtained by comparing known densities, using change-of-variables formulas, or exploiting known conditional laws. In discrete contexts, the derivative reduces to a simple ratio of point masses. In continuous contexts with smooth transformations, derivative computation often mirrors the structure of classical density transformations.

7.2 Handling null sets and almost-everywhere issues

Because uniqueness holds only \(\nu\)-almost everywhere, computed derivatives may be modified on \(\nu\)-null sets without changing the resulting measure. Practitioners typically choose convenient representatives, ensuring measurability and correct integral identities while ignoring behavior on sets that never affect \(\nu\)-integrals.

7.3 Numerical/algorithmic perspectives (high level)

When implementing measure changes numerically (e.g., in simulation or inference), one often approximates likelihood ratios or density estimates. The Radon–Nikodym derivative motivates weighting schemes, but computation may require regularization or careful treatment of regions where the derivative becomes large or unstable. High-level approaches include importance weighting, Monte Carlo reweighting, and density estimation followed by normalization.