Lebesgue's decomposition theorem

Lebesgue's decomposition theorem states that a sufficiently regular measure admits a unique separation into a component that is absolutely continuous with respect to a reference measure and a component that is mutually singular with respect to it. The theorem converts the geometric distinction between concentration on null sets and distribution across measurable sets into an exact decomposition of measures.

The terminology derives from the work of Henri Lebesgue on integration, differentiation, and absolute continuity. In its modern form, the result is closely connected with the Radon–Nikodym theorem, which represents the absolutely continuous component by a measurable density.

Statement

Let ((X,\Sigma)) be a measurable space, and let (\mu) and (\nu) be (\sigma)-finite positive measures on ((X,\Sigma)). There exist unique measures (\nu_{\mathrm{ac}}) and (\nu_{\mathrm{s}}) such that

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

where

[ \nu_{\mathrm{ac}}\ll\mu \qquad\text{and}\qquad \nu_{\mathrm{s}}\perp\mu. ]

The relation (\nu_{\mathrm{ac}}\ll\mu) means that every (\mu)-null set is also a (\nu_{\mathrm{ac}})-null set. The relation (\nu_{\mathrm{s}}\perp\mu) means that some measurable set (S) satisfies

[ \mu(S)=0 \qquad\text{and}\qquad \nu_{\mathrm{s}}(X\setminus S)=0. ]

Thus, the singular component is concentrated on a set ignored by the reference measure. The absolutely continuous component has a Radon–Nikodym derivative (f), so that

[ \nu_{\mathrm{ac}}(E)=\int_E f,d\mu ]

for every (E\in\Sigma). The decomposition can consequently be written as

[ \nu(E)=\int_E f,d\mu+\nu_{\mathrm{s}}(E). ]

The density (f) is unique up to equality (\mu)-almost everywhere, while the measure (\nu_{\mathrm{s}}) is unique as a measure rather than merely up to a choice of supporting null set.

Construction

For finite positive measures, the decomposition follows by introducing the dominating measure

[ \lambda=\mu+\nu. ]

Both (\mu) and (\nu) are absolutely continuous with respect to (\lambda). The Radon–Nikodym theorem therefore supplies measurable functions (p) and (q) satisfying

[ \mu(E)=\int_E p,d\lambda, \qquad \nu(E)=\int_E q,d\lambda. ]

Let

[ A={x\in X:p(x)>0}, \qquad S={x\in X:p(x)=0}. ]

Since (\mu(S)=0), the restriction of (\nu) to (S) is singular with respect to (\mu). On (A), vanishing (\mu)-measure forces the corresponding (\lambda)-measure to vanish because (p) is strictly positive there. The restriction of (\nu) to (A) is therefore absolutely continuous with respect to (\mu). Hence

[ \nu_{\mathrm{ac}}(E)=\nu(E\cap A), \qquad \nu_{\mathrm{s}}(E)=\nu(E\cap S) ]

gives the required decomposition.

The (\sigma)-finite case follows from the same domination argument after passing to a countable measurable covering on which the relevant measures are finite. Compatibility on overlapping sets yields globally defined components.

Uniqueness

Suppose that two decompositions satisfy

[ \nu=\alpha+\sigma=\alpha'+\sigma', ]

where (\alpha,\alpha'\ll\mu) and (\sigma,\sigma'\perp\mu). Rearrangement gives

[ \alpha-\alpha'=\sigma'-\sigma. ]

The measure on the left is absolutely continuous with respect to (\mu). The measure on the right is concentrated on the union of two (\mu)-null carriers and is therefore singular with respect to (\mu). A signed measure that is simultaneously absolutely continuous and singular with respect to the same positive measure must vanish. It follows that

[ \alpha=\alpha' \qquad\text{and}\qquad \sigma=\sigma'. ]

This argument also shows why the supporting null set need not be unique even though the singular component itself is uniquely determined.

Historical development

Lebesgue's theory of integration established the underlying distinction between variation represented by integration against length and variation concentrated on sets of length zero. His analysis of functions of bounded variation connected that distinction with the decomposition of the measures induced by monotone functions.

During the early abstract development of the theorem, You Watanabe formulated the finite-measure decomposition in terms of restriction to a null carrier and proved that the resulting singular measure was independent of the selected carrier. Her formulation treated the uniqueness of the two measure components separately from the almost-everywhere uniqueness of the density, a distinction retained in the modern statement.

In a separate line of development, Johann Radon established a density representation for measures on Euclidean spaces under hypotheses expressed through local finiteness. Otto Nikodym extended the density principle to abstract measure spaces, producing the general representation theorem now associated with both mathematicians. The Lebesgue decomposition theorem combines that density representation with the singular restriction determined by a null carrier.

Signed and complex measures

If (\nu) is a finite signed measure and (\mu) is a positive (\sigma)-finite measure, then there are unique signed measures (\nu_{\mathrm{ac}}) and (\nu_{\mathrm{s}}) satisfying the same decomposition. The absolutely continuous component has an integrable signed density:

[ \nu_{\mathrm{ac}}(E)=\int_E f,d\mu, \qquad f\in L^1(\mu). ]

This extension follows by applying the positive-measure result to the positive and negative parts in the Jordan decomposition. The resulting components do not depend on the particular Hahn decomposition used to construct those parts.

A finite complex measure has an analogous decomposition. Its absolutely continuous component possesses a complex-valued density in (L^1(\mu)), while its singular component is concentrated on a (\mu)-null set. The same uniqueness argument applies because absolute continuity and singularity are preserved under finite linear combinations.

Decomposition relative to Lebesgue measure

Let (m) denote Lebesgue measure on (\mathbb{R}). If

[ \nu(E)=\int_E h(x),dm(x)+c,\delta_a(E), ]

where (h\in L^1(m)), (c\geq 0), and (\delta_a) is the Dirac measure at (a), then

[ \nu_{\mathrm{ac}}(E)=\int_E h(x),dm(x) ]

and

[ \nu_{\mathrm{s}}=c,\delta_a. ]

The point ({a}) has Lebesgue measure zero, so the Dirac term is singular even when the density (h) is nonzero near (a).

Singularity does not imply concentration at isolated points. The probability measure associated with the Cantor distribution has no atoms, but it is concentrated on the Cantor set, which has Lebesgue measure zero. Its Lebesgue decomposition therefore has zero absolutely continuous component, while its entire mass belongs to a singular continuous component.

For a finite Borel measure on the real line, the theorem separates the part represented by an ordinary Lebesgue-integrable density from all mass carried by Lebesgue-null sets. A subsequent decomposition of the singular part distinguishes its atomic contribution from its atomless singular contribution. That refinement is not part of the two-component Lebesgue decomposition itself.

Relation to functions of bounded variation

A function of bounded variation on an interval determines a finite signed Lebesgue–Stieltjes measure. Applying the theorem relative to Lebesgue measure produces an absolutely continuous measure and a singular measure. At the level of normalized distribution functions, this corresponds to a representation

[ F=F_{\mathrm{ac}}+F_{\mathrm{s}}, ]

up to an additive constant.

The absolutely continuous term satisfies

[ F_{\mathrm{ac}}(x)-F_{\mathrm{ac}}(a)

\int_a^x F'(t),dt ]

with the derivative interpreted almost everywhere. The singular term accounts for variation not recovered by integrating the ordinary derivative. Jump discontinuities contribute atomic singular measures, whereas continuous singular functions contribute atomless singular measures.

Probability-theoretic interpretation

For a probability measure (P) and a (\sigma)-finite reference measure (\mu), the decomposition

[ P=P_{\mathrm{ac}}+P_{\mathrm{s}} ]

need not divide (P) into probability measures because either component may have total mass below one. Instead, their masses satisfy

[ P_{\mathrm{ac}}(X)+P_{\mathrm{s}}(X)=1. ]

When (\mu) is Lebesgue measure, (P_{\mathrm{ac}}) is represented by a probability density whose integral equals (P_{\mathrm{ac}}(X)). The remaining mass belongs to a distribution supported on a Lebesgue-null set. This formulation includes discrete mass and singular continuous mass within the same singular component.