Measure theory and analysis · existing Mathlib declaration

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

The theorem at a glance

Radon–Nikodym Theorem at a glance

A source-bound editorial overview of the exact Mathlib formulation. The image is explanatory; the statement map, formal type, and pinned source remain authoritative.

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.

Formal orientation

Statement map

This deterministic map orients the reader from the mathematical summary to the pinned declaration and exact checked form. The HTML statement below is authoritative.

Original-resolution inspection passed after copy cleanup and a semantic placement correction. The retained Scientific figure shows the null-set implication in the correct direction, nonnegative cellwise density over ν, aligned reconstruction of μ, the exact decomposition, and the singular part reduced to a zero baseline with its label directly attached. Only the eight permitted exact labels remain; the figure is a flat theorem-specific analysis diagram rather than a restyled concept schematic. — When no singular mass remains, the Radon–Nikodym density over ν reconstructs all of μ.

Complementary intuition

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.

The low-text schematic gives the theorem a visual identity; it does not replace the source-bound poster or exact statement.

Authoritative formal type

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

Selected evidence declaration text

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

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

These states distinguish upstream identity, local reproduction, review, and Atlas acceptance. This page is part of the public, read-only Mathlib landmark collection.

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.

Provenance

Origin and evidence stay separate

Existing declaration: MeasureTheory.Measure.absolutelyContinuous_iff_withDensity_rnDeriv_eq in mathlib
Relationship: The checked artifact is the upstream declaration itself
Preferred checked artifact: artifact.library.mathlib.radon-nikodym-theorem.v001
Source handling: Verified upstream reference; no mirrored source package