Documentation

Mathlib.MeasureTheory.Function.AEEqOfIntegral

From equality of integrals to equality of functions #

This file provides various statements of the general form "if two functions have the same integral on all sets, then they are equal almost everywhere". The different lemmas use various hypotheses on the class of functions, on the target space or on the possible finiteness of the measure.

Main statements #

All results listed below apply to two functions f, g, together with two main hypotheses,

For each of these results, we also provide a lemma about the equality of one function and 0. For example, Lp.ae_eq_zero_of_forall_set_integral_eq_zero.

We also register the corresponding lemma for integrals of ℝ≥0∞-valued functions, in ae_eq_of_forall_set_lintegral_eq_of_sigmaFinite.

Generally useful lemmas which are not related to integrals:

theorem MeasureTheory.ae_eq_zero_of_forall_inner {α : Type u_1} {E : Type u_2} {𝕜 : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [SecondCountableTopology E] {f : αE} (hf : ∀ (c : E), (fun (x : α) => c, f x⟫_𝕜) =ᶠ[MeasureTheory.Measure.ae μ] 0) :
theorem MeasureTheory.ae_eq_zero_of_forall_dual_of_isSeparable {α : Type u_1} {E : Type u_2} (𝕜 : Type u_3) {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [RCLike 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {t : Set E} (ht : TopologicalSpace.IsSeparable t) {f : αE} (hf : ∀ (c : NormedSpace.Dual 𝕜 E), (fun (x : α) => c (f x)) =ᶠ[MeasureTheory.Measure.ae μ] 0) (h't : ∀ᵐ (x : α) ∂μ, f x t) :
theorem MeasureTheory.ae_eq_zero_of_forall_dual {α : Type u_1} {E : Type u_2} (𝕜 : Type u_3) {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [RCLike 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [SecondCountableTopology E] {f : αE} (hf : ∀ (c : NormedSpace.Dual 𝕜 E), (fun (x : α) => c (f x)) =ᶠ[MeasureTheory.Measure.ae μ] 0) :
theorem MeasureTheory.ae_const_le_iff_forall_lt_measure_zero {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_3} [LinearOrder β] [TopologicalSpace β] [OrderTopology β] [FirstCountableTopology β] (f : αβ) (c : β) :
(∀ᵐ (x : α) ∂μ, c f x) b < c, μ {x : α | f x b} = 0
theorem MeasureTheory.ae_le_of_forall_set_lintegral_le_of_sigmaFinite {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] {f : αENNReal} {g : αENNReal} (hf : Measurable f) (hg : Measurable g) (h : ∀ (s : Set α), MeasurableSet sμ s < ∫⁻ (x : α) in s, f xμ ∫⁻ (x : α) in s, g xμ) :
theorem MeasureTheory.ae_le_of_forall_set_lintegral_le_of_sigmaFinite₀ {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] {f : αENNReal} {g : αENNReal} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) (h : ∀ (s : Set α), MeasurableSet sμ s < ∫⁻ (x : α) in s, f xμ ∫⁻ (x : α) in s, g xμ) :
theorem MeasureTheory.ae_eq_of_forall_set_lintegral_eq_of_sigmaFinite₀ {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] {f : αENNReal} {g : αENNReal} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) (h : ∀ (s : Set α), MeasurableSet sμ s < ∫⁻ (x : α) in s, f xμ = ∫⁻ (x : α) in s, g xμ) :
theorem MeasureTheory.ae_eq_of_forall_set_lintegral_eq_of_sigmaFinite {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] {f : αENNReal} {g : αENNReal} (hf : Measurable f) (hg : Measurable g) (h : ∀ (s : Set α), MeasurableSet sμ s < ∫⁻ (x : α) in s, f xμ = ∫⁻ (x : α) in s, g xμ) :
theorem MeasureTheory.ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α} (hfm : MeasureTheory.StronglyMeasurable f) (hf : MeasureTheory.Integrable f μ) (hf_zero : ∀ (s : Set α), MeasurableSet sμ s < 0 ∫ (x : α) in s, f xμ) :

Don't use this lemma. Use ae_nonneg_of_forall_set_integral_nonneg.

theorem MeasureTheory.ae_nonneg_of_forall_set_integral_nonneg {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α} (hf : MeasureTheory.Integrable f μ) (hf_zero : ∀ (s : Set α), MeasurableSet sμ s < 0 ∫ (x : α) in s, f xμ) :
theorem MeasureTheory.ae_le_of_forall_set_integral_le {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α} {g : α} (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ) (hf_le : ∀ (s : Set α), MeasurableSet sμ s < ∫ (x : α) in s, f xμ ∫ (x : α) in s, g xμ) :
theorem MeasureTheory.ae_nonneg_restrict_of_forall_set_integral_nonneg_inter {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α} {t : Set α} (hf : MeasureTheory.IntegrableOn f t μ) (hf_zero : ∀ (s : Set α), MeasurableSet sμ (s t) < 0 ∫ (x : α) in s t, f xμ) :
theorem MeasureTheory.ae_nonneg_of_forall_set_integral_nonneg_of_sigmaFinite {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] {f : α} (hf_int_finite : ∀ (s : Set α), MeasurableSet sμ s < MeasureTheory.IntegrableOn f s μ) (hf_zero : ∀ (s : Set α), MeasurableSet sμ s < 0 ∫ (x : α) in s, f xμ) :
theorem MeasureTheory.AEFinStronglyMeasurable.ae_nonneg_of_forall_set_integral_nonneg {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α} (hf : MeasureTheory.AEFinStronglyMeasurable f μ) (hf_int_finite : ∀ (s : Set α), MeasurableSet sμ s < MeasureTheory.IntegrableOn f s μ) (hf_zero : ∀ (s : Set α), MeasurableSet sμ s < 0 ∫ (x : α) in s, f xμ) :
theorem MeasureTheory.ae_nonneg_restrict_of_forall_set_integral_nonneg {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α} (hf_int_finite : ∀ (s : Set α), MeasurableSet sμ s < MeasureTheory.IntegrableOn f s μ) (hf_zero : ∀ (s : Set α), MeasurableSet sμ s < 0 ∫ (x : α) in s, f xμ) {t : Set α} (ht : MeasurableSet t) (hμt : μ t ) :
theorem MeasureTheory.ae_eq_zero_restrict_of_forall_set_integral_eq_zero_real {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α} (hf_int_finite : ∀ (s : Set α), MeasurableSet sμ s < MeasureTheory.IntegrableOn f s μ) (hf_zero : ∀ (s : Set α), MeasurableSet sμ s < ∫ (x : α) in s, f xμ = 0) {t : Set α} (ht : MeasurableSet t) (hμt : μ t ) :
theorem MeasureTheory.ae_eq_zero_restrict_of_forall_set_integral_eq_zero {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedSpace E] [CompleteSpace E] {f : αE} (hf_int_finite : ∀ (s : Set α), MeasurableSet sμ s < MeasureTheory.IntegrableOn f s μ) (hf_zero : ∀ (s : Set α), MeasurableSet sμ s < ∫ (x : α) in s, f xμ = 0) {t : Set α} (ht : MeasurableSet t) (hμt : μ t ) :
theorem MeasureTheory.ae_eq_restrict_of_forall_set_integral_eq {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedSpace E] [CompleteSpace E] {f : αE} {g : αE} (hf_int_finite : ∀ (s : Set α), MeasurableSet sμ s < MeasureTheory.IntegrableOn f s μ) (hg_int_finite : ∀ (s : Set α), MeasurableSet sμ s < MeasureTheory.IntegrableOn g s μ) (hfg_zero : ∀ (s : Set α), MeasurableSet sμ s < ∫ (x : α) in s, f xμ = ∫ (x : α) in s, g xμ) {t : Set α} (ht : MeasurableSet t) (hμt : μ t ) :
theorem MeasureTheory.ae_eq_zero_of_forall_set_integral_eq_of_sigmaFinite {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedSpace E] [CompleteSpace E] [MeasureTheory.SigmaFinite μ] {f : αE} (hf_int_finite : ∀ (s : Set α), MeasurableSet sμ s < MeasureTheory.IntegrableOn f s μ) (hf_zero : ∀ (s : Set α), MeasurableSet sμ s < ∫ (x : α) in s, f xμ = 0) :
theorem MeasureTheory.ae_eq_of_forall_set_integral_eq_of_sigmaFinite {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedSpace E] [CompleteSpace E] [MeasureTheory.SigmaFinite μ] {f : αE} {g : αE} (hf_int_finite : ∀ (s : Set α), MeasurableSet sμ s < MeasureTheory.IntegrableOn f s μ) (hg_int_finite : ∀ (s : Set α), MeasurableSet sμ s < MeasureTheory.IntegrableOn g s μ) (hfg_eq : ∀ (s : Set α), MeasurableSet sμ s < ∫ (x : α) in s, f xμ = ∫ (x : α) in s, g xμ) :
theorem MeasureTheory.AEFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedSpace E] [CompleteSpace E] {f : αE} (hf_int_finite : ∀ (s : Set α), MeasurableSet sμ s < MeasureTheory.IntegrableOn f s μ) (hf_zero : ∀ (s : Set α), MeasurableSet sμ s < ∫ (x : α) in s, f xμ = 0) (hf : MeasureTheory.AEFinStronglyMeasurable f μ) :
theorem MeasureTheory.AEFinStronglyMeasurable.ae_eq_of_forall_set_integral_eq {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedSpace E] [CompleteSpace E] {f : αE} {g : αE} (hf_int_finite : ∀ (s : Set α), MeasurableSet sμ s < MeasureTheory.IntegrableOn f s μ) (hg_int_finite : ∀ (s : Set α), MeasurableSet sμ s < MeasureTheory.IntegrableOn g s μ) (hfg_eq : ∀ (s : Set α), MeasurableSet sμ s < ∫ (x : α) in s, f xμ = ∫ (x : α) in s, g xμ) (hf : MeasureTheory.AEFinStronglyMeasurable f μ) (hg : MeasureTheory.AEFinStronglyMeasurable g μ) :
theorem MeasureTheory.Lp.ae_eq_zero_of_forall_set_integral_eq_zero {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedSpace E] [CompleteSpace E] {p : ENNReal} (f : (MeasureTheory.Lp E p μ)) (hp_ne_zero : p 0) (hp_ne_top : p ) (hf_int_finite : ∀ (s : Set α), MeasurableSet sμ s < MeasureTheory.IntegrableOn (f) s μ) (hf_zero : ∀ (s : Set α), MeasurableSet sμ s < ∫ (x : α) in s, f xμ = 0) :
theorem MeasureTheory.Lp.ae_eq_of_forall_set_integral_eq {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedSpace E] [CompleteSpace E] {p : ENNReal} (f : (MeasureTheory.Lp E p μ)) (g : (MeasureTheory.Lp E p μ)) (hp_ne_zero : p 0) (hp_ne_top : p ) (hf_int_finite : ∀ (s : Set α), MeasurableSet sμ s < MeasureTheory.IntegrableOn (f) s μ) (hg_int_finite : ∀ (s : Set α), MeasurableSet sμ s < MeasureTheory.IntegrableOn (g) s μ) (hfg : ∀ (s : Set α), MeasurableSet sμ s < ∫ (x : α) in s, f xμ = ∫ (x : α) in s, g xμ) :
theorem MeasureTheory.ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedSpace E] [CompleteSpace E] (hm : m m0) {f : αE} (hf_int_finite : ∀ (s : Set α), MeasurableSet sμ s < MeasureTheory.IntegrableOn f s μ) (hf_zero : ∀ (s : Set α), MeasurableSet sμ s < ∫ (x : α) in s, f xμ = 0) (hf : MeasureTheory.FinStronglyMeasurable f (MeasureTheory.Measure.trim μ hm)) :
theorem MeasureTheory.Integrable.ae_eq_zero_of_forall_set_integral_eq_zero {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedSpace E] [CompleteSpace E] {f : αE} (hf : MeasureTheory.Integrable f μ) (hf_zero : ∀ (s : Set α), MeasurableSet sμ s < ∫ (x : α) in s, f xμ = 0) :
theorem MeasureTheory.Integrable.ae_eq_of_forall_set_integral_eq {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedSpace E] [CompleteSpace E] (f : αE) (g : αE) (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ) (hfg : ∀ (s : Set α), MeasurableSet sμ s < ∫ (x : α) in s, f xμ = ∫ (x : α) in s, g xμ) :
theorem MeasureTheory.ae_eq_zero_of_forall_set_integral_isClosed_eq_zero {E : Type u_2} [NormedAddCommGroup E] [NormedSpace E] [CompleteSpace E] {β : Type u_3} [TopologicalSpace β] [MeasurableSpace β] [BorelSpace β] {μ : MeasureTheory.Measure β} {f : βE} (hf : MeasureTheory.Integrable f μ) (h'f : ∀ (s : Set β), IsClosed s∫ (x : β) in s, f xμ = 0) :

If an integrable function has zero integral on all closed sets, then it is zero almost everwhere.

theorem MeasureTheory.ae_eq_zero_of_forall_set_integral_isCompact_eq_zero {E : Type u_2} [NormedAddCommGroup E] [NormedSpace E] [CompleteSpace E] {β : Type u_3} [TopologicalSpace β] [MeasurableSpace β] [BorelSpace β] [SigmaCompactSpace β] [T2Space β] {μ : MeasureTheory.Measure β} {f : βE} (hf : MeasureTheory.Integrable f μ) (h'f : ∀ (s : Set β), IsCompact s∫ (x : β) in s, f xμ = 0) :

If an integrable function has zero integral on all compact sets in a sigma-compact space, then it is zero almost everwhere.

theorem MeasureTheory.ae_eq_zero_of_forall_set_integral_isCompact_eq_zero' {E : Type u_2} [NormedAddCommGroup E] [NormedSpace E] [CompleteSpace E] {β : Type u_3} [TopologicalSpace β] [MeasurableSpace β] [BorelSpace β] [SigmaCompactSpace β] [T2Space β] {μ : MeasureTheory.Measure β} {f : βE} (hf : MeasureTheory.LocallyIntegrable f μ) (h'f : ∀ (s : Set β), IsCompact s∫ (x : β) in s, f xμ = 0) :

If a locally integrable function has zero integral on all compact sets in a sigma-compact space, then it is zero almost everwhere.

theorem MeasureTheory.AEMeasurable.ae_eq_of_forall_set_lintegral_eq {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : αENNReal} {g : αENNReal} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) (hfi : ∫⁻ (x : α), f xμ ) (hgi : ∫⁻ (x : α), g xμ ) (hfg : ∀ ⦃s : Set α⦄, MeasurableSet sμ s < ∫⁻ (x : α) in s, f xμ = ∫⁻ (x : α) in s, g xμ) :
theorem MeasureTheory.withDensity_eq_iff {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : αENNReal} {g : αENNReal} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) (hfi : ∫⁻ (x : α), f xμ ) :