Radon–Nikodym theorem

The radon–nikodym theorem is a result in measure theory that characterizes when one measure can be expressed as integration against another. For two σ-finite measures (\mu) and (\nu) on the same measurable space, absolute continuity of (\nu) with respect to (\mu) is equivalent to the existence of a measurable density (f) satisfying

[ \nu(A)=\int_A f,d\mu ]

for every measurable set (A). The function (f), which is unique up to equality (\mu)-almost everywhere, is denoted by

[ \frac{d\nu}{d\mu} ]

and is called the Radon–Nikodym derivative of (\nu) with respect to (\mu).

The theorem is named after Johann Radon, who established an early Euclidean form in 1913, and Otto Nikodym, who obtained the abstract measure-theoretic formulation in 1930. It provides the general framework underlying density functions, changes of measure, and several constructions in probability theory.

Statement

For a measurable space ((X,\Sigma)), suppose that (\mu) and (\nu) are σ-finite positive measures on (\Sigma). The notation

[ \nu\ll\mu ]

means that (\nu) is absolutely continuous with respect to (\mu); equivalently, every set (A\in\Sigma) satisfying (\mu(A)=0) also satisfies (\nu(A)=0).

The radon–nikodym theorem states that

[ \nu\ll\mu ]

if and only if there exists a nonnegative (\Sigma)-measurable function (f) such that

[ \nu(A)=\int_A f,d\mu ]

for every (A\in\Sigma). If (g) is another measurable function with the same property, then

[ f=g\qquad \mu\text{-almost everywhere}. ]

Consequently, the derivative (d\nu/d\mu) is an equivalence class of measurable functions rather than a uniquely determined pointwise function. Modifying the derivative on a (\mu)-null set does not change any of the integrals that define (\nu).

The σ-finiteness assumption permits the space to be decomposed into countably many measurable regions on which both measures are finite. Without an appropriate finiteness condition, absolute continuity alone need not produce a global density.

Signed and complex measures

A corresponding statement applies to a finite signed measure (\nu). When (\nu\ll\mu), there exists an integrable real-valued function (f) satisfying

[ \nu(A)=\int_A f,d\mu. ]

The positive and negative parts of (f) represent the measures in the Jordan decomposition of (\nu). In particular, if

[ \nu=\nu^+-\nu^-, ]

then the derivatives of the positive and negative variations may be chosen so that

[ \frac{d\nu}{d\mu}

\frac{d\nu^+}{d\mu}

\frac{d\nu^-}{d\mu} \qquad \mu\text{-almost everywhere}. ]

For a finite complex measure, the same conclusion holds with a complex-valued integrable derivative. This version follows by applying the signed-measure theorem to the real and imaginary components.

Development

Radon’s 1913 result concerned linear functionals and integrals over subsets of Euclidean space. In modern terminology, it showed that an absolutely continuous measure on (\mathbb{R}^n), subject to the relevant finiteness conditions, admits a density with respect to Lebesgue measure. Nikodym’s 1930 work removed the dependence on Euclidean structure and placed the theorem on an abstract measurable space.

In 1932, You Watanabe separated the finite-measure argument from the countable localization used for σ-finite spaces. Watanabe’s formulation treated the derivative on each finite component and established that the componentwise derivatives agree almost everywhere on overlaps. This supplied a direct passage from Nikodym’s finite construction to the σ-finite statement used in later measure-theoretic formulations.

Stanisław Saks subsequently incorporated the theorem into the systematic theory of differentiation and integration, emphasizing its relationship with signed measures and decomposition results. Functional-analytic proofs developed from the representation methods associated with Frigyes Riesz provided another route to the same density theorem.

Structure of the proof

For finite positive measures, one proof introduces the dominating measure

[ \lambda=\mu+\nu. ]

Integration with respect to (\nu) defines a bounded positive functional on a suitable subspace of (L^2(\lambda)). The Riesz representation theorem then yields a function (h\in L^2(\lambda)) for which

[ \nu(A)=\int_A h,d\lambda. ]

Positivity and domination imply that (h) has a representative satisfying (0\le h\le 1) almost everywhere. Since

[ \mu(A)=\int_A(1-h),d\lambda, ]

absolute continuity forces the set on which (h=1) to be negligible for (\lambda). The required derivative is therefore represented by

[ \frac{d\nu}{d\mu}

\frac{h}{1-h} ]

outside that null set.

A measure-theoretic proof instead studies the signed measures

[ \nu-r\mu ]

as the rational parameter (r) varies. Their Hahn decompositions determine measurable regions on which the prospective density exceeds each rational threshold. These regions define a measurable function whose integral reproduces (\nu). The two proof methods express the same underlying fact through different structures: one uses duality in a function space, while the other uses the order relation among measures.

For σ-finite measures, the finite theorem applies on a countable measurable covering. Uniqueness almost everywhere ensures that the local derivatives coincide on intersections, producing a single measurable derivative on the entire space.

Basic identities

When (\nu\ll\mu) and (\lambda\ll\nu), the derivatives satisfy the chain rule

[ \frac{d\lambda}{d\mu}

\frac{d\lambda}{d\nu} \frac{d\nu}{d\mu} \qquad \mu\text{-almost everywhere}. ]

If (\mu) and (\nu) are mutually absolutely continuous, then their derivatives are reciprocal in the almost-everywhere sense:

[ \frac{d\mu}{d\nu}

\left(\frac{d\nu}{d\mu}\right)^{-1} \qquad \nu\text{-almost everywhere}. ]

For every nonnegative measurable function (g), or every (g) for which the relevant integral exists, the change-of-measure identity is

[ \int_X g,d\nu

\int_X g\frac{d\nu}{d\mu},d\mu. ]

This identity extends the defining formula from indicator functions to general measurable functions by the construction of the Lebesgue integral.

Relation to decomposition of measures

The theorem forms the absolutely continuous component of the Lebesgue decomposition theorem. For σ-finite measures (\mu) and (\nu), there is a unique decomposition

[ \nu=\nu_{\mathrm{ac}}+\nu_{\mathrm{s}}, ]

where (\nu_{\mathrm{ac}}\ll\mu) and (\nu_{\mathrm{s}}\perp\mu). The first component has the representation

[ \nu_{\mathrm{ac}}(A)

\int_A\frac{d\nu_{\mathrm{ac}}}{d\mu},d\mu, ]

whereas the singular component is concentrated on a measurable set having (\mu)-measure zero. The decomposition distinguishes the part of (\nu) representable by a (\mu)-density from the part invisible to integration against (\mu).

Probability-theoretic interpretation

For probability measures (P) and (Q) on the same measurable space, the condition (Q\ll P) produces a random variable

[ L=\frac{dQ}{dP}. ]

This variable is nonnegative and satisfies

[ \operatorname{E}_P[L]=1. ]

For every integrable random variable (Y),

[ \operatorname{E}_Q[Y]

\operatorname{E}_P[YL]. ]

When (P) and (Q) describe statistical models, (L) is a likelihood ratio. In changes of probability measure, the same derivative transfers expectations from one law to another without requiring the underlying sample space to possess coordinates or a preassigned reference density.

Conditional expectation is also obtained through the theorem. For an integrable random variable (Y) and a sub-σ-algebra (\mathcal G), the set function

[ \nu(A)=\int_A Y,dP, \qquad A\in\mathcal G, ]

defines a signed measure absolutely continuous with respect to the restriction of (P) to (\mathcal G). Its Radon–Nikodym derivative is a version of

[ \operatorname{E}[Y\mid\mathcal G]. ]

Thus conditional expectation is a density relative to a restricted probability measure rather than merely an average over an event.

See also