Existing in Mathlib · Analysis
Tonelli's Theorem
For measurable spaces α and β, measures μ and ν with ν s-finite, and an ℝ≥0∞-valued function f that is almost everywhere measurable for μ.prod ν, integrating f over the product measure equals first integrating f(x, y) over y against ν and then integrating those fiber totals over x against μ.
- product measures
- Lebesgue integration
- nonnegative functions
- iterated integrals
- almost-everywhere measurability
- s-finite measures
Exact theorem
Exact Mathlib statement
theorem MeasureTheory.lintegral_prod {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] {μ : Measure α} {ν : Measure β} [SFinite ν] (f : α × β → ℝ≥0∞) (hf : AEMeasurable f (μ.prod ν)) : ∫⁻ z, f z ∂μ.prod ν = ∫⁻ x, ∫⁻ y, f (x, y) ∂ν ∂μThe theorem at a glance
Tonelli's theorem at a glance
A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.
Tonelli's theorem at a glance
Detailed visual description
The poster begins with the product-space measure and an extended-nonnegative height field, separates the field into measurable y-fibers indexed by x, and recombines their nonnegative masses into the same total. Its exact-scope footer retains the AEMeasurable hypothesis, [SFinite ν], the selected order of integration, and the possibility of infinite mass while excluding signed-integral Fubini.
Statement structure
From hypotheses to conclusion
This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.
Product mass resolved into measurable fibers
Detailed visual description
An entirely above-base product-space surface flows through representative parallel curtains into corresponding ordinary trough profiles. Forward-only gold guides carry those profiles into one continuous outer basin. The finite displayed family is illustrative rather than a claim that the measured family is finite, and the plain lower gold bridge compares the product and iterated constructions without medals, scales, symbols, or text.
Why it matters
A mathematical landmark
Tonelli's theorem is the fundamental nonnegative product-integration principle behind repeated integration, product probability spaces, and much of modern measure theory. Mathlib's declaration records the exact extended-nonnegative and almost-everywhere measurable form on an s-finite inner measure.
ProofAtlas record
What has been checked
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 product-measure lintegral form of Tonelli's theorem. With measurable spaces α and β, measures μ and ν, [SFinite ν], and an ℝ≥0∞-valued f that is almost everywhere measurable for μ.prod ν, the product-space lintegral equals the iterated lintegral that integrates y against ν and then x against μ. The common value may be infinite. This selected endpoint is not the signed-integral Fubini theorem, does not itself exchange the two integration orders, and does not omit its measurability or SFinite hypothesis.
- The selected declaration is for ℝ≥0∞-valued lintegrals; it is not the signed- or Bochner-integral Fubini theorem.
- The conclusion uses the fixed order ∫⁻ x, ∫⁻ y with ν inside and μ outside; the selected declaration does not itself assert equality with the reverse order.
- The theorem retains an AEMeasurable hypothesis for f with respect to μ.prod ν and an ambient [SFinite ν] instance.
- The common extended-nonnegative value is allowed to be ∞; no finiteness or integrability conclusion is asserted.
- ProofAtlas is indexing an existing Mathlib theorem, and generated explanations and visuals are not proof evidence.
Source and local evidence
Where the theorem comes from
- Existing declaration
MeasureTheory.lintegral_prodin mathlib- Relationship
- The checked artifact is the upstream declaration itself
- Local Lean evidence
artifact.library.mathlib.tonelli-theorem.v001