Mathlib theorem · Existing formal mathematics
ProofAtlas

Existing in Mathlib · Measure theory and analysis

Hölder's Inequality

For two almost-everywhere measurable nonnegative extended-real functions, conjugate powers control the lower integral of their pointwise product by the product of two separate power-integral scales.

Exact theorem

Exact Mathlib statement

theorem ENNReal.lintegral_mul_le_Lp_mul_Lq {α : Type*} [MeasurableSpace α] (μ : Measure α) {p q : ℝ} (hpq : p.HolderConjugate q) {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) : (∫⁻ a, (f * g) a ∂μ) ≤ (∫⁻ a, f a ^ p ∂μ) ^ (1 / p) * (∫⁻ a, g a ^ q ∂μ) ^ (1 / q)

The theorem at a glance

Hölder's inequality for lintegrals at a glance

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

The product lintegral stays below the product of the two conjugate power-integral scales. Explanatory diagram.
Detailed visual description

Two independently AEMeasurable ℝ≥0∞-valued functions share one measured domain. Their pointwise product feeds the left side of Hölder's inequality, while separate p- and q-power lintegrals supply the two multiplied scales on the right. The checked route splits off vanishing and top-valued cases, then normalizes the finite nonzero branch, applies Young's inequality pointwise, integrates, and rescales.

Statement structure

From hypotheses to conclusion

Statement map for Hölder's InequalityConjugate power-integral scales bound the lower integral of the pointwise product of two AEMeasurable ℝ≥0∞-valued functions. The pinned upstream declaration is ENNReal.lintegral_mul_le_Lp_mul_Lq. The exact checked statement is theorem ENNReal.lintegral_mul_le_Lp_mul_Lq {α : Type*} [MeasurableSpace α] (μ : Measure α) {p q : ℝ} (hpq : p.HolderConjugate q) {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) : (∫⁻ a, (f * g) a ∂μ) ≤ (∫⁻ a, f a ^ p ∂μ) ^ (1 / p) * (∫⁻ a, g a ^ q ∂μ) ^ (1 / q).Mathematical readingConjugate power-integralscales bound the lowerintegral of thepointwise product of twoAEMeasurable ℝ≥0∞-valuedfunctions.Pinned declarationmathlib ·ENNReal.lintegral_mul_le_Lp_mul_LqExact checked formtheoremENNReal.lintegral_mul_le_Lp_mul_Lq{α : Type*}[MeasurableSpace α] (μ :Measure α) {p q : ℝ}(hpq : p.HolderConjugateq) {f g : α → ℝ≥0∞} (hf: AEMeasurable f μ) (hg: AEMeasurable g μ) :(∫⁻ a, (f * g) a ∂μ) ≤(∫⁻ a, f a ^ p ∂μ) ^ (1/ p) * (∫⁻ a, g a ^ q∂μ) ^ (1 / q)Statement map for Hölder's InequalityConjugate power-integral scales bound the lower integral of the pointwise product of two AEMeasurable ℝ≥0∞-valued functions. The pinned upstream declaration is ENNReal.lintegral_mul_le_Lp_mul_Lq. The exact checked statement is theorem ENNReal.lintegral_mul_le_Lp_mul_Lq {α : Type*} [MeasurableSpace α] (μ : Measure α) {p q : ℝ} (hpq : p.HolderConjugate q) {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) : (∫⁻ a, (f * g) a ∂μ) ≤ (∫⁻ a, f a ^ p ∂μ) ^ (1 / p) * (∫⁻ a, g a ^ q ∂μ) ^ (1 / q).Mathematical readingConjugate power-integralscales bound the lowerintegral of thepointwise product of twoAEMeasurable ℝ≥0∞-valuedfunctions.Pinned declarationmathlib ·ENNReal.lintegral_mul_le_Lp_mul_LqExact checked formtheoremENNReal.lintegral_mul_le_Lp_mul_Lq{α : Type*}[MeasurableSpace α] (μ :Measure α) {p q : ℝ}(hpq : p.HolderConjugateq) {f g : α → ℝ≥0∞} (hf: AEMeasurable f μ) (hg: AEMeasurable g μ) :(∫⁻ a, (f * g) a ∂μ) ≤(∫⁻ a, f a ^ p ∂μ) ^ (1/ p) * (∫⁻ a, g a ^ q∂μ) ^ (1 / q)

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

Separate conjugate power scales jointly control the accumulated pointwise product. Explanatory scientific diagram.
Detailed visual description

Two nonnegative profiles share one measured domain and are simultaneously positive on a common central interval. Their pointwise product vanishes outside that overlap and accumulates below a paired support fed by the two separate power-scale vessels. The construction communicates the upper bound without asserting equality, an optimizer, finite integrals, or a probability-only specialization.

Why it matters

A mathematical landmark

Hölder's inequality is one of the central bridges between multiplication and integrability, underpinning duality, norm estimates, and much of modern analysis. Mathlib's selected declaration is especially instructive because its extended-real form keeps zero and infinite cases explicit while the finite branch exposes the normalization-and-Young-inequality mechanism.

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 Hölder inequality for two AEMeasurable functions f and g from a measured type α to the extended nonnegative reals ℝ≥0∞. For Holder-conjugate real exponents p and q, it bounds the lower Lebesgue integral of the pointwise product by the product of the corresponding p- and q-power lintegral scales. The declaration includes zero and top-valued cases; it does not state the signed real- or complex-valued absolute-value form, assume finite power integrals, characterize equality, or assert existence or uniqueness of an optimizer.

Source and local evidence

Where the theorem comes from

Existing declaration
ENNReal.lintegral_mul_le_Lp_mul_Lq in mathlib
Relationship
The checked artifact is the upstream declaration itself
Local Lean evidence
artifact.library.mathlib.holder-inequality.v001
Source
Open the pinned upstream reference