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.

absolute continuity of measures
Radon–Nikodym derivative
measure density
Lebesgue decomposition
ENNReal-valued functions

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.

An ivory analysis poster layers a forest-green measure ν, an antique-gold Radon–Nikodym density, and the reconstructed cobalt measure μ above the exact biconditional and positive-measure scope. — Assuming the recorded Lebesgue decomposition, absolute continuity is equivalent to exact reconstruction by the Radon–Nikodym density.

Statement structure

From hypotheses to conclusion

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

A null-set implication leads to a nonnegative density over ν that reconstructs μ, while the singular component is reduced to a zero baseline. — When no singular mass remains, the Radon–Nikodym density over ν reconstructs all of μ.

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.

Proof Atlas did not originate the Radon–Nikodym theorem or Mathlib's declaration.
The selected declaration requires HaveLebesgueDecomposition μ ν and concerns positive measures with an ENNReal-valued density, not the signed- or vector-measure variants.
The generated explanation and visuals are not proof evidence.

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