File: Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean

full_name: MeasureTheory.ae_eq_zero_of_forall_inner
code: theorem ae_eq_zero_of_forall_inner [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
    [SecondCountableTopology E] {f : α → E} (hf : ∀ c : E, (fun x => (inner c (f x) : 𝕜)) =ᵐ[μ] 0) :
    f =ᵐ[μ] 0 
goal: f =ᵐ[μ] 0

full_name: MeasureTheory.«_aux_Mathlib_MeasureTheory_Function_AEEqOfIntegral___macroRules_MeasureTheory_term⟪_,_⟫_1»
code: local notation "⟪" x ", " y "⟫" => y x
goal: Lean.Macro

full_name: MeasureTheory.ae_eq_zero_of_forall_dual_of_isSeparable
code: theorem ae_eq_zero_of_forall_dual_of_isSeparable [NormedAddCommGroup E] [NormedSpace 𝕜 E]
    {t : Set E} (ht : TopologicalSpace.IsSeparable t) {f : α → E}
    (hf : ∀ c : Dual 𝕜 E, (fun x => ⟪f x, c⟫) =ᵐ[μ] 0) (h't : ∀ᵐ x ∂μ, f x ∈ t) : f =ᵐ[μ] 0 
goal: f =ᵐ[μ] 0

full_name: MeasureTheory.ae_eq_zero_of_forall_dual
code: theorem ae_eq_zero_of_forall_dual [NormedAddCommGroup E] [NormedSpace 𝕜 E]
    [SecondCountableTopology E] {f : α → E} (hf : ∀ c : Dual 𝕜 E, (fun x => ⟪f x, c⟫) =ᵐ[μ] 0) :
    f =ᵐ[μ] 0 
goal: f =ᵐ[μ] 0

full_name: MeasureTheory.ae_const_le_iff_forall_lt_measure_zero
code: theorem ae_const_le_iff_forall_lt_measure_zero {β} [LinearOrder β] [TopologicalSpace β]
    [OrderTopology β] [FirstCountableTopology β] (f : α → β) (c : β) :
    (∀ᵐ x ∂μ, c ≤ f x) ↔ ∀ b < c, μ {x | f x ≤ b} = 0 
goal: (∀ᵐ (x : α) ∂μ, c ≤ f x) ↔ ∀ b < c, μ {x | f x ≤ b} = 0

full_name: MeasureTheory.ae_le_of_forall_setLIntegral_le_of_sigmaFinite₀
code: theorem ae_le_of_forall_setLIntegral_le_of_sigmaFinite₀ [SigmaFinite μ]
    {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ)
    (h : ∀ s, MeasurableSet s → μ s < ∞ → ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x in s, g x ∂μ) :
    f ≤ᵐ[μ] g 
goal: f ≤ᵐ[μ] g

full_name: MeasureTheory.ae_le_of_forall_setLIntegral_le_of_sigmaFinite
code: theorem ae_le_of_forall_setLIntegral_le_of_sigmaFinite [SigmaFinite μ] {f g : α → ℝ≥0∞}
    (hf : Measurable f)
    (h : ∀ s, MeasurableSet s → μ s < ∞ → (∫⁻ x in s, f x ∂μ) ≤ ∫⁻ x in s, g x ∂μ) : f ≤ᵐ[μ] g 
goal: f ≤ᵐ[μ] g

full_name: MeasureTheory.ae_eq_of_forall_setLIntegral_eq_of_sigmaFinite₀
code: theorem ae_eq_of_forall_setLIntegral_eq_of_sigmaFinite₀ [SigmaFinite μ]
    {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
    (h : ∀ s, MeasurableSet s → μ s < ∞ → ∫⁻ x in s, f x ∂μ = ∫⁻ x in s, g x ∂μ) : f =ᵐ[μ] g 
goal: f =ᵐ[μ] g

full_name: MeasureTheory.ae_eq_of_forall_setLIntegral_eq_of_sigmaFinite
code: theorem ae_eq_of_forall_setLIntegral_eq_of_sigmaFinite [SigmaFinite μ] {f g : α → ℝ≥0∞}
    (hf : Measurable f) (hg : Measurable g)
    (h : ∀ s, MeasurableSet s → μ s < ∞ → ∫⁻ x in s, f x ∂μ = ∫⁻ x in s, g x ∂μ) : f =ᵐ[μ] g 
goal: f =ᵐ[μ] g

full_name: MeasureTheory.ae_nonneg_of_forall_setIntegral_nonneg
code: theorem ae_nonneg_of_forall_setIntegral_nonneg (hf : Integrable f μ)
    (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) : 0 ≤ᵐ[μ] f 
goal: 0 ≤ᵐ[μ] f

full_name: MeasureTheory.ae_le_of_forall_setIntegral_le
code: theorem ae_le_of_forall_setIntegral_le {f g : α → ℝ} (hf : Integrable f μ) (hg : Integrable g μ)
    (hf_le : ∀ s, MeasurableSet s → μ s < ∞ → (∫ x in s, f x ∂μ) ≤ ∫ x in s, g x ∂μ) :
    f ≤ᵐ[μ] g 
goal: f ≤ᵐ[μ] g

full_name: MeasureTheory.ae_nonneg_restrict_of_forall_setIntegral_nonneg_inter
code: theorem ae_nonneg_restrict_of_forall_setIntegral_nonneg_inter {f : α → ℝ} {t : Set α}
    (hf : IntegrableOn f t μ)
    (hf_zero : ∀ s, MeasurableSet s → μ (s ∩ t) < ∞ → 0 ≤ ∫ x in s ∩ t, f x ∂μ) :
    0 ≤ᵐ[μ.restrict t] f 
goal: 0 ≤ᵐ[μ.restrict t] f

full_name: MeasureTheory.ae_nonneg_of_forall_setIntegral_nonneg_of_sigmaFinite
code: theorem ae_nonneg_of_forall_setIntegral_nonneg_of_sigmaFinite [SigmaFinite μ] {f : α → ℝ}
    (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
    (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) : 0 ≤ᵐ[μ] f 
goal: 0 ≤ᵐ[μ] f

full_name: MeasureTheory.AEFinStronglyMeasurable.ae_nonneg_of_forall_setIntegral_nonneg
code: theorem AEFinStronglyMeasurable.ae_nonneg_of_forall_setIntegral_nonneg {f : α → ℝ}
    (hf : AEFinStronglyMeasurable f μ)
    (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
    (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) : 0 ≤ᵐ[μ] f 
goal: 0 ≤ᵐ[μ] f

full_name: MeasureTheory.ae_nonneg_restrict_of_forall_setIntegral_nonneg
code: theorem ae_nonneg_restrict_of_forall_setIntegral_nonneg {f : α → ℝ}
    (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
    (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) {t : Set α}
    (ht : MeasurableSet t) (hμt : μ t ≠ ∞) : 0 ≤ᵐ[μ.restrict t] f 
goal: 0 ≤ᵐ[μ.restrict t] f

full_name: MeasureTheory.ae_eq_zero_restrict_of_forall_setIntegral_eq_zero_real
code: theorem ae_eq_zero_restrict_of_forall_setIntegral_eq_zero_real {f : α → ℝ}
    (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
    (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) {t : Set α}
    (ht : MeasurableSet t) (hμt : μ t ≠ ∞) : f =ᵐ[μ.restrict t] 0 
goal: f =ᵐ[μ.restrict t] 0

full_name: MeasureTheory.ae_eq_zero_restrict_of_forall_setIntegral_eq_zero
code: theorem ae_eq_zero_restrict_of_forall_setIntegral_eq_zero {f : α → E}
    (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
    (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) {t : Set α}
    (ht : MeasurableSet t) (hμt : μ t ≠ ∞) : f =ᵐ[μ.restrict t] 0 
goal: f =ᵐ[μ.restrict t] 0

full_name: MeasureTheory.ae_eq_restrict_of_forall_setIntegral_eq
code: theorem ae_eq_restrict_of_forall_setIntegral_eq {f g : α → E}
    (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
    (hg_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn g s μ)
    (hfg_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ)
    {t : Set α} (ht : MeasurableSet t) (hμt : μ t ≠ ∞) : f =ᵐ[μ.restrict t] g 
goal: f =ᵐ[μ.restrict t] g

full_name: MeasureTheory.ae_eq_zero_of_forall_setIntegral_eq_of_sigmaFinite
code: theorem ae_eq_zero_of_forall_setIntegral_eq_of_sigmaFinite [SigmaFinite μ] {f : α → E}
    (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
    (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) : f =ᵐ[μ] 0 
goal: f =ᵐ[μ] 0

full_name: MeasureTheory.ae_eq_of_forall_setIntegral_eq_of_sigmaFinite
code: theorem ae_eq_of_forall_setIntegral_eq_of_sigmaFinite [SigmaFinite μ] {f g : α → E}
    (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
    (hg_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn g s μ)
    (hfg_eq : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ) :
    f =ᵐ[μ] g 
goal: f =ᵐ[μ] g

full_name: MeasureTheory.AEFinStronglyMeasurable.ae_eq_zero_of_forall_setIntegral_eq_zero
code: theorem AEFinStronglyMeasurable.ae_eq_zero_of_forall_setIntegral_eq_zero {f : α → E}
    (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
    (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0)
    (hf : AEFinStronglyMeasurable f μ) : f =ᵐ[μ] 0 
goal: f =ᵐ[μ] 0

full_name: MeasureTheory.AEFinStronglyMeasurable.ae_eq_of_forall_setIntegral_eq
code: theorem AEFinStronglyMeasurable.ae_eq_of_forall_setIntegral_eq {f g : α → E}
    (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
    (hg_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn g s μ)
    (hfg_eq : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ)
    (hf : AEFinStronglyMeasurable f μ) (hg : AEFinStronglyMeasurable g μ) : f =ᵐ[μ] g 
goal: f =ᵐ[μ] g

full_name: MeasureTheory.Lp.ae_eq_zero_of_forall_setIntegral_eq_zero
code: theorem Lp.ae_eq_zero_of_forall_setIntegral_eq_zero (f : Lp E p μ) (hp_ne_zero : p ≠ 0)
    (hp_ne_top : p ≠ ∞) (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
    (hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) : f =ᵐ[μ] 0 
goal: ↑↑f =ᵐ[μ] 0

full_name: MeasureTheory.Lp.ae_eq_of_forall_setIntegral_eq
code: theorem Lp.ae_eq_of_forall_setIntegral_eq (f g : Lp E p μ) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
    (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
    (hg_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn g s μ)
    (hfg : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ) :
    f =ᵐ[μ] g 
goal: ↑↑f =ᵐ[μ] ↑↑g

full_name: MeasureTheory.ae_eq_zero_of_forall_setIntegral_eq_of_finStronglyMeasurable_trim
code: theorem ae_eq_zero_of_forall_setIntegral_eq_of_finStronglyMeasurable_trim (hm : m ≤ m0) {f : α → E}
    (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn f s μ)
    (hf_zero : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, f x ∂μ = 0)
    (hf : FinStronglyMeasurable f (μ.trim hm)) : f =ᵐ[μ] 0 
goal: f =ᵐ[μ] 0

full_name: MeasureTheory.Integrable.ae_eq_zero_of_forall_setIntegral_eq_zero
code: theorem Integrable.ae_eq_zero_of_forall_setIntegral_eq_zero {f : α → E} (hf : Integrable f μ)
    (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) : f =ᵐ[μ] 0 
goal: f =ᵐ[μ] 0

full_name: MeasureTheory.Integrable.ae_eq_of_forall_setIntegral_eq
code: theorem Integrable.ae_eq_of_forall_setIntegral_eq (f g : α → E) (hf : Integrable f μ)
    (hg : Integrable g μ)
    (hfg : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ) :
    f =ᵐ[μ] g 
goal: f =ᵐ[μ] g

full_name: MeasureTheory.ae_eq_zero_of_forall_setIntegral_isClosed_eq_zero
code: /-- If an integrable function has zero integral on all closed sets, then it is zero
almost everwhere. -/
lemma ae_eq_zero_of_forall_setIntegral_isClosed_eq_zero {μ : Measure β} {f : β → E}
    (hf : Integrable f μ) (h'f : ∀ (s : Set β), IsClosed s → ∫ x in s, f x ∂μ = 0) :
    f =ᵐ[μ] 0 
goal: f =ᵐ[μ] 0

full_name: MeasureTheory.ae_eq_zero_of_forall_setIntegral_isCompact_eq_zero
code: /-- If an integrable function has zero integral on all compact sets in a sigma-compact space, then
it is zero almost everwhere. -/
lemma ae_eq_zero_of_forall_setIntegral_isCompact_eq_zero
    [SigmaCompactSpace β] [R1Space β] {μ : Measure β} {f : β → E} (hf : Integrable f μ)
    (h'f : ∀ (s : Set β), IsCompact s → ∫ x in s, f x ∂μ = 0) :
    f =ᵐ[μ] 0 
goal: f =ᵐ[μ] 0

full_name: MeasureTheory.ae_eq_zero_of_forall_setIntegral_isCompact_eq_zero'
code: /-- If a locally integrable function has zero integral on all compact sets in a sigma-compact space,
then it is zero almost everwhere. -/
lemma ae_eq_zero_of_forall_setIntegral_isCompact_eq_zero'
    [SigmaCompactSpace β] [R1Space β] {μ : Measure β} {f : β → E} (hf : LocallyIntegrable f μ)
    (h'f : ∀ (s : Set β), IsCompact s → ∫ x in s, f x ∂μ = 0) :
    f =ᵐ[μ] 0 
goal: f =ᵐ[μ] 0

full_name: MeasureTheory.AEMeasurable.ae_eq_of_forall_setLIntegral_eq
code: theorem AEMeasurable.ae_eq_of_forall_setLIntegral_eq {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ)
    (hg : AEMeasurable g μ) (hfi : ∫⁻ x, f x ∂μ ≠ ∞) (hgi : ∫⁻ x, g x ∂μ ≠ ∞)
    (hfg : ∀ ⦃s⦄, MeasurableSet s → μ s < ∞ → ∫⁻ x in s, f x ∂μ = ∫⁻ x in s, g x ∂μ) :
    f =ᵐ[μ] g 
goal: f =ᵐ[μ] g

full_name: MeasureTheory.withDensity_eq_iff_of_sigmaFinite
code: theorem withDensity_eq_iff_of_sigmaFinite [SigmaFinite μ] {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ)
    (hg : AEMeasurable g μ) : μ.withDensity f = μ.withDensity g ↔ f =ᵐ[μ] g 
goal: μ.withDensity f = μ.withDensity g ↔ f =ᵐ[μ] g

full_name: MeasureTheory.withDensity_eq_iff
code: theorem withDensity_eq_iff {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ)
    (hg : AEMeasurable g μ) (hfi : ∫⁻ x, f x ∂μ ≠ ∞) :
    μ.withDensity f = μ.withDensity g ↔ f =ᵐ[μ] g 
goal: μ.withDensity f = μ.withDensity g ↔ f =ᵐ[μ] g