Mathlib theorem · Existing formal mathematics
ProofAtlas

Existing in Mathlib · Measure theory and analysis

Radon–Nikodym Theorem

When μ assigns zero mass wherever ν does, the Radon–Nikodym derivative acts as a nonnegative density that reweights ν to recover all of μ exactly—and the reconstruction equality also implies that absolute continuity.

Exact theorem

Exact Mathlib statement

theorem MeasureTheory.Measure.absolutelyContinuous_iff_withDensity_rnDeriv_eq {α : Type*} {m : MeasurableSpace α} {μ ν : Measure α} [HaveLebesgueDecomposition μ ν] : μ ≪ ν ↔ ν.withDensity (rnDeriv μ ν) = μ

The theorem at a glance

Radon–Nikodym Theorem at a glance

A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.

Assuming the recorded Lebesgue decomposition, absolute continuity is equivalent to exact reconstruction by the Radon–Nikodym density. Explanatory diagram.
Detailed visual description

The poster presents the assumption HaveLebesgueDecomposition μ ν, then aligns a green ν base, a variable gold density layer, and a blue μ field. The central row states μ ≪ ν if and only if ν.withDensity (rnDeriv μ ν) = μ. A muted red singular fragment dissolves before a footer limits the result to positive measures and excludes signed- and vector-measure claims.

Statement structure

Statement and scope

Statement map for Radon–Nikodym TheoremUnder the required Lebesgue decomposition, μ is absolutely continuous with respect to ν exactly when weighting ν by rnDeriv μ ν reconstructs μ. Claim boundary: This page indexes Mathlib's positive-measure Radon–Nikodym theorem. For measures μ and ν on one measurable space, it assumes HaveLebesgueDecomposition μ ν and states the exact equivalence μ ≪ ν ↔ ν.withDensity (rnDeriv μ ν) = μ. The density is ENNReal-valued. The selected declaration is not the signed-measure theorem, not a vector-measure theorem, and not an assumption-free existence claim for arbitrary measure pairs. The pinned upstream declaration is MeasureTheory.Measure.absolutelyContinuous_iff_withDensity_rnDeriv_eq. The complete formal statement is available in the Exact Mathlib statement section.Mathematical readingUnder the requiredLebesgue decomposition,μ is absolutelycontinuous with respectto ν exactly whenweighting ν by rnDeriv μν reconstructs μ.Claim boundaryThis page indexesMathlib'spositive-measureRadon–Nikodym theorem.For measures μ and ν onone measurable space, itassumesHaveLebesgueDecompositionμ ν and states the exactequivalence μ ≪ ν ↔ν.withDensity (rnDeriv μν) = μ. The density isENNReal-valued. Theselected declaration isnot the signed-measuretheorem, not avector-measure theorem,and not anassumption-freeexistence claim forarbitrary measure pairs.Pinned declarationmathlib ·MeasureTheory.Measure.absolutelyContinuous_iff_withDensity_rnDeriv_eqStatement map for Radon–Nikodym TheoremUnder the required Lebesgue decomposition, μ is absolutely continuous with respect to ν exactly when weighting ν by rnDeriv μ ν reconstructs μ. Claim boundary: This page indexes Mathlib's positive-measure Radon–Nikodym theorem. For measures μ and ν on one measurable space, it assumes HaveLebesgueDecomposition μ ν and states the exact equivalence μ ≪ ν ↔ ν.withDensity (rnDeriv μ ν) = μ. The density is ENNReal-valued. The selected declaration is not the signed-measure theorem, not a vector-measure theorem, and not an assumption-free existence claim for arbitrary measure pairs. The pinned upstream declaration is MeasureTheory.Measure.absolutelyContinuous_iff_withDensity_rnDeriv_eq. The complete formal statement is available in the Exact Mathlib statement section.Mathematical readingUnder the requiredLebesgue decomposition,μ is absolutelycontinuous with respectto ν exactly whenweighting ν by rnDeriv μν reconstructs μ.Claim boundaryThis page indexesMathlib'spositive-measureRadon–Nikodym theorem.For measures μ and ν onone measurable space, itassumesHaveLebesgueDecompositionμ ν and states the exactequivalence μ ≪ ν ↔ν.withDensity (rnDeriv μν) = μ. The density isENNReal-valued. Theselected declaration isnot the signed-measuretheorem, not avector-measure theorem,and not anassumption-freeexistence claim forarbitrary measure pairs.Pinned declarationmathlib ·MeasureTheory.Measure.absolutelyContinuous_iff_withDensity_rnDeriv_eq

Read the exact Mathlib declaration

This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.

With a Lebesgue decomposition, absolute continuity is equivalent to reconstructing μ by weighting ν with its Radon–Nikodym density. Explanatory scientific diagram.
Detailed visual description

The top row has three outlined panels with dashed grids and dashed set outlines. Blue reference bars occupy the left panel; gold density bars stand over a blue base in the middle; dark reconstructed bars appear on the right. Dashed arrows connect the middle and right panels. Two smaller rows below show red, gold, and dark contribution strips; the lower-left singular strip is a dashed zero baseline, contrasting with the nonzero red strip above it. The four small labels are reference, density, result, and singular. The bar heights are qualitative schematic marks, not measured data or a numerical equality plot. The exact adjacent theorem concerns positive measures with a Lebesgue decomposition and an ENNReal-valued density; it is not an assumption-free, signed-measure, or vector-measure statement.

Why it matters

A mathematical landmark

The Radon–Nikodym theorem is the central representation principle for absolutely continuous measures. It turns a qualitative relation between null sets into an exact density formula and underlies foundational constructions throughout analysis and probability.

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally
Locally reproducedExact upstream declaration replayed
Reviewed pageCurrent public presentation reviewed
Accepted Atlas resultNot recorded for the preferred artifact

Mathlib is the source of the theorem; the local Lean replay and page review are separate.

Claim boundary

No new theorem is claimed

This page indexes Mathlib's positive-measure Radon–Nikodym theorem. For measures μ and ν on one measurable space, it assumes HaveLebesgueDecomposition μ ν and states the exact equivalence μ ≪ ν ↔ ν.withDensity (rnDeriv μ ν) = μ. The density is ENNReal-valued. The selected declaration is not the signed-measure theorem, not a vector-measure theorem, and not an assumption-free existence claim for arbitrary measure pairs.

Source and local evidence

Where the theorem comes from

Existing declaration
MeasureTheory.Measure.absolutelyContinuous_iff_withDensity_rnDeriv_eq in mathlib
Relationship
The checked artifact is the upstream declaration itself
Local Lean evidence
artifact.library.mathlib.radon-nikodym-theorem.v001
Source
Open the pinned upstream reference