Lebesgue decomposition theorem

The Lebesgue decomposition theorem states that a σ-finite measure splits uniquely into a part absolutely continuous with respect to another σ-finite measure and a part mutually singular with it.

Absolute continuity of measures: A measure is absolutely continuous with respect to another when every null set of the latter is also null for the former. This property identifies the component that can be represented by a density.
Measure space: A measure space consists of a set, a σ-algebra of measurable subsets, and a measure assigning them nonnegative sizes. The theorem is formulated for measures defined on a shared measurable space.
Probability measure: A probability measure assigns total mass one to the outcomes of a measurable space. Probability laws can be decomposed relative to a chosen reference distribution.
Equivalent measures: Two measures are equivalent when each is absolutely continuous with respect to the other. Equivalence links measures through shared null sets, unlike the decomposition’s singular component.
Mutually singular measures: Two measures are mutually singular when they are concentrated on disjoint measurable sets. This relation distinguishes the component supported apart from the reference measure.
σ-finite measure: A measure is σ-finite if its space is a countable union of measurable sets, each having finite measure. σ-finiteness is the hypothesis that enables the theorem’s standard existence and uniqueness result.
Probability density function: A probability density function is a nonnegative measurable function whose integral defines probabilities relative to a reference measure. The absolutely continuous part of a probability law has a density of this kind.
Dominated convergence theorem: The dominated convergence theorem gives conditions under which limits pass through integrals. Despite the shared name, it concerns convergence of integrals rather than splitting measures.
Radon–Nikodym theorem: The Radon–Nikodym theorem represents an absolutely continuous σ-finite measure as an integral of a measurable density. It gives the density description of the theorem’s absolutely continuous component.
Null set: A null set is a measurable set assigned measure zero by a specified measure. Absolute continuity and singularity are both expressed through sets that have measure zero.

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F. and M. Riesz theoremRelated: The result rules out the singular part when the Fourier coefficients vanish on one side.