File: Mathlib/MeasureTheory/Integral/IntegrableOn.lean

full_name: StronglyMeasurableAtFilter
code: /-- A function `f` is strongly measurable at a filter `l` w.r.t. a measure `μ` if it is
ae strongly measurable w.r.t. `μ.restrict s` for some `s ∈ l`. -/
def StronglyMeasurableAtFilter (f : α → β) (l : Filter α) (μ : Measure α := by volume_tac) 
goal: Prop

full_name: stronglyMeasurableAt_bot
code: @[simp]
theorem stronglyMeasurableAt_bot {f : α → β} : StronglyMeasurableAtFilter f ⊥ μ 
goal: StronglyMeasurableAtFilter f ⊥ μ

full_name: StronglyMeasurableAtFilter.eventually
code: protected theorem StronglyMeasurableAtFilter.eventually (h : StronglyMeasurableAtFilter f l μ) :
    ∀ᶠ s in l.smallSets, AEStronglyMeasurable f (μ.restrict s) 
goal: ∀ᶠ (s : Set α) in l.smallSets, MeasureTheory.AEStronglyMeasurable f (μ.restrict s)

full_name: StronglyMeasurableAtFilter.filter_mono
code: protected theorem StronglyMeasurableAtFilter.filter_mono (h : StronglyMeasurableAtFilter f l μ)
    (h' : l' ≤ l) : StronglyMeasurableAtFilter f l' μ 
goal: StronglyMeasurableAtFilter f l' μ

full_name: MeasureTheory.AEStronglyMeasurable.stronglyMeasurableAtFilter
code: protected theorem MeasureTheory.AEStronglyMeasurable.stronglyMeasurableAtFilter
    (h : AEStronglyMeasurable f μ) : StronglyMeasurableAtFilter f l μ 
goal: StronglyMeasurableAtFilter f l μ

full_name: AeStronglyMeasurable.stronglyMeasurableAtFilter_of_mem
code: theorem AeStronglyMeasurable.stronglyMeasurableAtFilter_of_mem {s}
    (h : AEStronglyMeasurable f (μ.restrict s)) (hl : s ∈ l) : StronglyMeasurableAtFilter f l μ 
goal: StronglyMeasurableAtFilter f l μ

full_name: MeasureTheory.StronglyMeasurable.stronglyMeasurableAtFilter
code: protected theorem MeasureTheory.StronglyMeasurable.stronglyMeasurableAtFilter
    (h : StronglyMeasurable f) : StronglyMeasurableAtFilter f l μ 
goal: StronglyMeasurableAtFilter f l μ

full_name: MeasureTheory.hasFiniteIntegral_restrict_of_bounded
code: theorem hasFiniteIntegral_restrict_of_bounded [NormedAddCommGroup E] {f : α → E} {s : Set α}
    {μ : Measure α} {C} (hs : μ s < ∞) (hf : ∀ᵐ x ∂μ.restrict s, ‖f x‖ ≤ C) :
    HasFiniteIntegral f (μ.restrict s) 
goal: MeasureTheory.HasFiniteIntegral f (μ.restrict s)

full_name: MeasureTheory.IntegrableOn
code: /-- A function is `IntegrableOn` a set `s` if it is almost everywhere strongly measurable on `s`
and if the integral of its pointwise norm over `s` is less than infinity. -/
def IntegrableOn (f : α → E) (s : Set α) (μ : Measure α := by volume_tac) : Prop 
goal: Prop

full_name: MeasureTheory.IntegrableOn.integrable
code: theorem IntegrableOn.integrable (h : IntegrableOn f s μ) : Integrable f (μ.restrict s) 
goal: MeasureTheory.Integrable f (μ.restrict s)

full_name: MeasureTheory.integrableOn_empty
code: @[simp]
theorem integrableOn_empty : IntegrableOn f ∅ μ 
goal: MeasureTheory.IntegrableOn f ∅ μ

full_name: MeasureTheory.integrableOn_univ
code: @[simp]
theorem integrableOn_univ : IntegrableOn f univ μ ↔ Integrable f μ 
goal: MeasureTheory.IntegrableOn f Set.univ μ ↔ MeasureTheory.Integrable f μ

full_name: MeasureTheory.integrableOn_zero
code: theorem integrableOn_zero : IntegrableOn (fun _ => (0 : E)) s μ 
goal: MeasureTheory.IntegrableOn (fun x => 0) s μ

full_name: MeasureTheory.integrableOn_const
code: @[simp]
theorem integrableOn_const {C : E} : IntegrableOn (fun _ => C) s μ ↔ C = 0 ∨ μ s < ∞ 
goal: MeasureTheory.IntegrableOn (fun x => C) s μ ↔ C = 0 ∨ μ s < ⊤

full_name: MeasureTheory.IntegrableOn.mono
code: theorem IntegrableOn.mono (h : IntegrableOn f t ν) (hs : s ⊆ t) (hμ : μ ≤ ν) : IntegrableOn f s μ 
goal: MeasureTheory.IntegrableOn f s μ

full_name: MeasureTheory.IntegrableOn.mono_set
code: theorem IntegrableOn.mono_set (h : IntegrableOn f t μ) (hst : s ⊆ t) : IntegrableOn f s μ 
goal: MeasureTheory.IntegrableOn f s μ

full_name: MeasureTheory.IntegrableOn.mono_measure
code: theorem IntegrableOn.mono_measure (h : IntegrableOn f s ν) (hμ : μ ≤ ν) : IntegrableOn f s μ 
goal: MeasureTheory.IntegrableOn f s μ

full_name: MeasureTheory.IntegrableOn.mono_set_ae
code: theorem IntegrableOn.mono_set_ae (h : IntegrableOn f t μ) (hst : s ≤ᵐ[μ] t) : IntegrableOn f s μ 
goal: MeasureTheory.IntegrableOn f s μ

full_name: MeasureTheory.IntegrableOn.congr_set_ae
code: theorem IntegrableOn.congr_set_ae (h : IntegrableOn f t μ) (hst : s =ᵐ[μ] t) : IntegrableOn f s μ 
goal: MeasureTheory.IntegrableOn f s μ

full_name: MeasureTheory.IntegrableOn.congr_fun_ae
code: theorem IntegrableOn.congr_fun_ae (h : IntegrableOn f s μ) (hst : f =ᵐ[μ.restrict s] g) :
    IntegrableOn g s μ 
goal: MeasureTheory.IntegrableOn g s μ

full_name: MeasureTheory.integrableOn_congr_fun_ae
code: theorem integrableOn_congr_fun_ae (hst : f =ᵐ[μ.restrict s] g) :
    IntegrableOn f s μ ↔ IntegrableOn g s μ 
goal: MeasureTheory.IntegrableOn f s μ ↔ MeasureTheory.IntegrableOn g s μ

full_name: MeasureTheory.IntegrableOn.congr_fun
code: theorem IntegrableOn.congr_fun (h : IntegrableOn f s μ) (hst : EqOn f g s) (hs : MeasurableSet s) :
    IntegrableOn g s μ 
goal: MeasureTheory.IntegrableOn g s μ

full_name: MeasureTheory.integrableOn_congr_fun
code: theorem integrableOn_congr_fun (hst : EqOn f g s) (hs : MeasurableSet s) :
    IntegrableOn f s μ ↔ IntegrableOn g s μ 
goal: MeasureTheory.IntegrableOn f s μ ↔ MeasureTheory.IntegrableOn g s μ

full_name: MeasureTheory.Integrable.integrableOn
code: theorem Integrable.integrableOn (h : Integrable f μ) : IntegrableOn f s μ 
goal: MeasureTheory.IntegrableOn f s μ

full_name: MeasureTheory.IntegrableOn.restrict
code: theorem IntegrableOn.restrict (h : IntegrableOn f s μ) (hs : MeasurableSet s) :
    IntegrableOn f s (μ.restrict t) 
goal: MeasureTheory.IntegrableOn f s (μ.restrict t)

full_name: MeasureTheory.IntegrableOn.inter_of_restrict
code: theorem IntegrableOn.inter_of_restrict (h : IntegrableOn f s (μ.restrict t)) :
    IntegrableOn f (s ∩ t) μ 
goal: MeasureTheory.IntegrableOn f (s ∩ t) μ

full_name: MeasureTheory.Integrable.piecewise
code: lemma Integrable.piecewise [DecidablePred (· ∈ s)]
    (hs : MeasurableSet s) (hf : IntegrableOn f s μ) (hg : IntegrableOn g sᶜ μ) :
    Integrable (s.piecewise f g) μ 
goal: MeasureTheory.Integrable (s.piecewise f g) μ

full_name: MeasureTheory.IntegrableOn.left_of_union
code: theorem IntegrableOn.left_of_union (h : IntegrableOn f (s ∪ t) μ) : IntegrableOn f s μ 
goal: MeasureTheory.IntegrableOn f s μ

full_name: MeasureTheory.IntegrableOn.right_of_union
code: theorem IntegrableOn.right_of_union (h : IntegrableOn f (s ∪ t) μ) : IntegrableOn f t μ 
goal: MeasureTheory.IntegrableOn f t μ

full_name: MeasureTheory.IntegrableOn.union
code: theorem IntegrableOn.union (hs : IntegrableOn f s μ) (ht : IntegrableOn f t μ) :
    IntegrableOn f (s ∪ t) μ 
goal: MeasureTheory.IntegrableOn f (s ∪ t) μ

full_name: MeasureTheory.integrableOn_union
code: @[simp]
theorem integrableOn_union : IntegrableOn f (s ∪ t) μ ↔ IntegrableOn f s μ ∧ IntegrableOn f t μ 
goal: MeasureTheory.IntegrableOn f (s ∪ t) μ ↔ MeasureTheory.IntegrableOn f s μ ∧ MeasureTheory.IntegrableOn f t μ

full_name: MeasureTheory.integrableOn_singleton_iff
code: @[simp]
theorem integrableOn_singleton_iff {x : α} [MeasurableSingletonClass α] :
    IntegrableOn f {x} μ ↔ f x = 0 ∨ μ {x} < ∞ 
goal: MeasureTheory.IntegrableOn f {x} μ ↔ f x = 0 ∨ μ {x} < ⊤

full_name: MeasureTheory.integrableOn_finite_biUnion
code: @[simp]
theorem integrableOn_finite_biUnion {s : Set β} (hs : s.Finite) {t : β → Set α} :
    IntegrableOn f (⋃ i ∈ s, t i) μ ↔ ∀ i ∈ s, IntegrableOn f (t i) μ 
goal: MeasureTheory.IntegrableOn f (⋃ i ∈ s, t i) μ ↔ ∀ i ∈ s, MeasureTheory.IntegrableOn f (t i) μ

full_name: MeasureTheory.integrableOn_finset_iUnion
code: @[simp]
theorem integrableOn_finset_iUnion {s : Finset β} {t : β → Set α} :
    IntegrableOn f (⋃ i ∈ s, t i) μ ↔ ∀ i ∈ s, IntegrableOn f (t i) μ 
goal: MeasureTheory.IntegrableOn f (⋃ i ∈ s, t i) μ ↔ ∀ i ∈ s, MeasureTheory.IntegrableOn f (t i) μ

full_name: MeasureTheory.integrableOn_finite_iUnion
code: @[simp]
theorem integrableOn_finite_iUnion [Finite β] {t : β → Set α} :
    IntegrableOn f (⋃ i, t i) μ ↔ ∀ i, IntegrableOn f (t i) μ 
goal: MeasureTheory.IntegrableOn f (⋃ i, t i) μ ↔ ∀ (i : β), MeasureTheory.IntegrableOn f (t i) μ

full_name: MeasureTheory.IntegrableOn.add_measure
code: theorem IntegrableOn.add_measure (hμ : IntegrableOn f s μ) (hν : IntegrableOn f s ν) :
    IntegrableOn f s (μ + ν) 
goal: MeasureTheory.IntegrableOn f s (μ + ν)

full_name: MeasureTheory.integrableOn_add_measure
code: @[simp]
theorem integrableOn_add_measure :
    IntegrableOn f s (μ + ν) ↔ IntegrableOn f s μ ∧ IntegrableOn f s ν 
goal: MeasureTheory.IntegrableOn f s (μ + ν) ↔ MeasureTheory.IntegrableOn f s μ ∧ MeasureTheory.IntegrableOn f s ν

full_name: MeasurableEmbedding.integrableOn_map_iff
code: theorem _root_.MeasurableEmbedding.integrableOn_map_iff [MeasurableSpace β] {e : α → β}
    (he : MeasurableEmbedding e) {f : β → E} {μ : Measure α} {s : Set β} :
    IntegrableOn f s (μ.map e) ↔ IntegrableOn (f ∘ e) (e ⁻¹' s) μ 
goal: MeasureTheory.IntegrableOn f s (MeasureTheory.Measure.map e μ) ↔ MeasureTheory.IntegrableOn (f ∘ e) (e ⁻¹' s) μ

full_name: MeasurableEmbedding.integrableOn_iff_comap
code: theorem _root_.MeasurableEmbedding.integrableOn_iff_comap [MeasurableSpace β] {e : α → β}
    (he : MeasurableEmbedding e) {f : β → E} {μ : Measure β} {s : Set β} (hs : s ⊆ range e) :
    IntegrableOn f s μ ↔ IntegrableOn (f ∘ e) (e ⁻¹' s) (μ.comap e) 
goal: MeasureTheory.IntegrableOn f s μ ↔ MeasureTheory.IntegrableOn (f ∘ e) (e ⁻¹' s) (MeasureTheory.Measure.comap e μ)

full_name: MeasureTheory.integrableOn_map_equiv
code: theorem integrableOn_map_equiv [MeasurableSpace β] (e : α ≃ᵐ β) {f : β → E} {μ : Measure α}
    {s : Set β} : IntegrableOn f s (μ.map e) ↔ IntegrableOn (f ∘ e) (e ⁻¹' s) μ 
goal: MeasureTheory.IntegrableOn f s (MeasureTheory.Measure.map (⇑e) μ) ↔ MeasureTheory.IntegrableOn (f ∘ ⇑e) (⇑e ⁻¹' s) μ

full_name: MeasureTheory.MeasurePreserving.integrableOn_comp_preimage
code: theorem MeasurePreserving.integrableOn_comp_preimage [MeasurableSpace β] {e : α → β} {ν}
    (h₁ : MeasurePreserving e μ ν) (h₂ : MeasurableEmbedding e) {f : β → E} {s : Set β} :
    IntegrableOn (f ∘ e) (e ⁻¹' s) μ ↔ IntegrableOn f s ν 
goal: MeasureTheory.IntegrableOn (f ∘ e) (e ⁻¹' s) μ ↔ MeasureTheory.IntegrableOn f s ν

full_name: MeasureTheory.MeasurePreserving.integrableOn_image
code: theorem MeasurePreserving.integrableOn_image [MeasurableSpace β] {e : α → β} {ν}
    (h₁ : MeasurePreserving e μ ν) (h₂ : MeasurableEmbedding e) {f : β → E} {s : Set α} :
    IntegrableOn f (e '' s) ν ↔ IntegrableOn (f ∘ e) s μ 
goal: MeasureTheory.IntegrableOn f (e '' s) ν ↔ MeasureTheory.IntegrableOn (f ∘ e) s μ

full_name: MeasureTheory.integrable_indicator_iff
code: theorem integrable_indicator_iff (hs : MeasurableSet s) :
    Integrable (indicator s f) μ ↔ IntegrableOn f s μ 
goal: MeasureTheory.Integrable (s.indicator f) μ ↔ MeasureTheory.IntegrableOn f s μ

full_name: MeasureTheory.IntegrableOn.integrable_indicator
code: theorem IntegrableOn.integrable_indicator (h : IntegrableOn f s μ) (hs : MeasurableSet s) :
    Integrable (indicator s f) μ 
goal: MeasureTheory.Integrable (s.indicator f) μ

full_name: MeasureTheory.Integrable.indicator
code: theorem Integrable.indicator (h : Integrable f μ) (hs : MeasurableSet s) :
    Integrable (indicator s f) μ 
goal: MeasureTheory.Integrable (s.indicator f) μ

full_name: MeasureTheory.IntegrableOn.indicator
code: theorem IntegrableOn.indicator (h : IntegrableOn f s μ) (ht : MeasurableSet t) :
    IntegrableOn (indicator t f) s μ 
goal: MeasureTheory.IntegrableOn (t.indicator f) s μ

full_name: MeasureTheory.integrable_indicatorConstLp
code: theorem integrable_indicatorConstLp {E} [NormedAddCommGroup E] {p : ℝ≥0∞} {s : Set α}
    (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (c : E) :
    Integrable (indicatorConstLp p hs hμs c) μ 
goal: MeasureTheory.Integrable (↑↑(MeasureTheory.indicatorConstLp p hs hμs c)) μ

full_name: MeasureTheory.IntegrableOn.restrict_toMeasurable
code: /-- If a function is integrable on a set `s` and nonzero there, then the measurable hull of `s` is
well behaved: the restriction of the measure to `toMeasurable μ s` coincides with its restriction
to `s`. -/
theorem IntegrableOn.restrict_toMeasurable (hf : IntegrableOn f s μ) (h's : ∀ x ∈ s, f x ≠ 0) :
    μ.restrict (toMeasurable μ s) = μ.restrict s 
goal: μ.restrict (MeasureTheory.toMeasurable μ s) = μ.restrict s

full_name: MeasureTheory.IntegrableOn.of_ae_diff_eq_zero
code: /-- If a function is integrable on a set `s`, and vanishes on `t \ s`, then it is integrable on `t`
if `t` is null-measurable. -/
theorem IntegrableOn.of_ae_diff_eq_zero (hf : IntegrableOn f s μ) (ht : NullMeasurableSet t μ)
    (h't : ∀ᵐ x ∂μ, x ∈ t \ s → f x = 0) : IntegrableOn f t μ 
goal: MeasureTheory.IntegrableOn f t μ

full_name: MeasureTheory.IntegrableOn.of_forall_diff_eq_zero
code: /-- If a function is integrable on a set `s`, and vanishes on `t \ s`, then it is integrable on `t`
if `t` is measurable. -/
theorem IntegrableOn.of_forall_diff_eq_zero (hf : IntegrableOn f s μ) (ht : MeasurableSet t)
    (h't : ∀ x ∈ t \ s, f x = 0) : IntegrableOn f t μ 
goal: MeasureTheory.IntegrableOn f t μ

full_name: MeasureTheory.IntegrableOn.integrable_of_ae_not_mem_eq_zero
code: /-- If a function is integrable on a set `s` and vanishes almost everywhere on its complement,
then it is integrable. -/
theorem IntegrableOn.integrable_of_ae_not_mem_eq_zero (hf : IntegrableOn f s μ)
    (h't : ∀ᵐ x ∂μ, x ∉ s → f x = 0) : Integrable f μ 
goal: MeasureTheory.Integrable f μ

full_name: MeasureTheory.IntegrableOn.integrable_of_forall_not_mem_eq_zero
code: /-- If a function is integrable on a set `s` and vanishes everywhere on its complement,
then it is integrable. -/
theorem IntegrableOn.integrable_of_forall_not_mem_eq_zero (hf : IntegrableOn f s μ)
    (h't : ∀ x, x ∉ s → f x = 0) : Integrable f μ 
goal: MeasureTheory.Integrable f μ

full_name: MeasureTheory.integrableOn_iff_integrable_of_support_subset
code: theorem integrableOn_iff_integrable_of_support_subset (h1s : support f ⊆ s) :
    IntegrableOn f s μ ↔ Integrable f μ 
goal: MeasureTheory.IntegrableOn f s μ ↔ MeasureTheory.Integrable f μ

full_name: MeasureTheory.integrableOn_Lp_of_measure_ne_top
code: theorem integrableOn_Lp_of_measure_ne_top {E} [NormedAddCommGroup E] {p : ℝ≥0∞} {s : Set α}
    (f : Lp E p μ) (hp : 1 ≤ p) (hμs : μ s ≠ ∞) : IntegrableOn f s μ 
goal: MeasureTheory.IntegrableOn (↑↑f) s μ

full_name: MeasureTheory.Integrable.lintegral_lt_top
code: theorem Integrable.lintegral_lt_top {f : α → ℝ} (hf : Integrable f μ) :
    (∫⁻ x, ENNReal.ofReal (f x) ∂μ) < ∞ 
goal: ∫⁻ (x : α), ENNReal.ofReal (f x) ∂μ < ⊤

full_name: MeasureTheory.IntegrableOn.setLIntegral_lt_top
code: theorem IntegrableOn.setLIntegral_lt_top {f : α → ℝ} {s : Set α} (hf : IntegrableOn f s μ) :
    (∫⁻ x in s, ENNReal.ofReal (f x) ∂μ) < ∞ 
goal: ∫⁻ (x : α) in s, ENNReal.ofReal (f x) ∂μ < ⊤

full_name: MeasureTheory.IntegrableAtFilter
code: /-- We say that a function `f` is *integrable at filter* `l` if it is integrable on some
set `s ∈ l`. Equivalently, it is eventually integrable on `s` in `l.smallSets`. -/
def IntegrableAtFilter (f : α → E) (l : Filter α) (μ : Measure α := by volume_tac) 
goal: Prop

full_name: MeasurableEmbedding.integrableAtFilter_map_iff
code: theorem _root_.MeasurableEmbedding.integrableAtFilter_map_iff [MeasurableSpace β] {e : α → β}
    (he : MeasurableEmbedding e) {f : β → E} :
    IntegrableAtFilter f (l.map e) (μ.map e) ↔ IntegrableAtFilter (f ∘ e) l μ 
goal: MeasureTheory.IntegrableAtFilter f (Filter.map e l) (MeasureTheory.Measure.map e μ) ↔
  MeasureTheory.IntegrableAtFilter (f ∘ e) l μ

full_name: MeasurableEmbedding.integrableAtFilter_iff_comap
code: theorem _root_.MeasurableEmbedding.integrableAtFilter_iff_comap [MeasurableSpace β] {e : α → β}
    (he : MeasurableEmbedding e) {f : β → E} {μ : Measure β} :
    IntegrableAtFilter f (l.map e) μ ↔ IntegrableAtFilter (f ∘ e) l (μ.comap e) 
goal: MeasureTheory.IntegrableAtFilter f (Filter.map e l) μ ↔
  MeasureTheory.IntegrableAtFilter (f ∘ e) l (MeasureTheory.Measure.comap e μ)

full_name: MeasureTheory.Integrable.integrableAtFilter
code: theorem Integrable.integrableAtFilter (h : Integrable f μ) (l : Filter α) :
    IntegrableAtFilter f l μ 
goal: MeasureTheory.IntegrableAtFilter f l μ

full_name: MeasureTheory.IntegrableAtFilter.eventually
code: protected theorem IntegrableAtFilter.eventually (h : IntegrableAtFilter f l μ) :
    ∀ᶠ s in l.smallSets, IntegrableOn f s μ 
goal: ∀ᶠ (s : Set α) in l.smallSets, MeasureTheory.IntegrableOn f s μ

full_name: MeasureTheory.IntegrableAtFilter.add
code: protected theorem IntegrableAtFilter.add {f g : α → E}
    (hf : IntegrableAtFilter f l μ) (hg : IntegrableAtFilter g l μ) :
    IntegrableAtFilter (f + g) l μ 
goal: MeasureTheory.IntegrableAtFilter (f + g) l μ

full_name: MeasureTheory.IntegrableAtFilter.neg
code: protected theorem IntegrableAtFilter.neg {f : α → E} (hf : IntegrableAtFilter f l μ) :
    IntegrableAtFilter (-f) l μ 
goal: MeasureTheory.IntegrableAtFilter (-f) l μ

full_name: MeasureTheory.IntegrableAtFilter.sub
code: protected theorem IntegrableAtFilter.sub {f g : α → E}
    (hf : IntegrableAtFilter f l μ) (hg : IntegrableAtFilter g l μ) :
    IntegrableAtFilter (f - g) l μ 
goal: MeasureTheory.IntegrableAtFilter (f - g) l μ

full_name: MeasureTheory.IntegrableAtFilter.smul
code: protected theorem IntegrableAtFilter.smul {𝕜 : Type*} [NormedAddCommGroup 𝕜] [SMulZeroClass 𝕜 E]
    [BoundedSMul 𝕜 E] {f : α → E} (hf : IntegrableAtFilter f l μ) (c : 𝕜) :
    IntegrableAtFilter (c • f) l μ 
goal: MeasureTheory.IntegrableAtFilter (c • f) l μ

full_name: MeasureTheory.IntegrableAtFilter.norm
code: protected theorem IntegrableAtFilter.norm (hf : IntegrableAtFilter f l μ) :
    IntegrableAtFilter (fun x => ‖f x‖) l μ 
goal: MeasureTheory.IntegrableAtFilter (fun x => ‖f x‖) l μ

full_name: MeasureTheory.IntegrableAtFilter.filter_mono
code: theorem IntegrableAtFilter.filter_mono (hl : l ≤ l') (hl' : IntegrableAtFilter f l' μ) :
    IntegrableAtFilter f l μ 
goal: MeasureTheory.IntegrableAtFilter f l μ

full_name: MeasureTheory.IntegrableAtFilter.inf_of_left
code: theorem IntegrableAtFilter.inf_of_left (hl : IntegrableAtFilter f l μ) :
    IntegrableAtFilter f (l ⊓ l') μ 
goal: MeasureTheory.IntegrableAtFilter f (l ⊓ l') μ

full_name: MeasureTheory.IntegrableAtFilter.inf_of_right
code: theorem IntegrableAtFilter.inf_of_right (hl : IntegrableAtFilter f l μ) :
    IntegrableAtFilter f (l' ⊓ l) μ 
goal: MeasureTheory.IntegrableAtFilter f (l' ⊓ l) μ

full_name: MeasureTheory.IntegrableAtFilter.inf_ae_iff
code: @[simp]
theorem IntegrableAtFilter.inf_ae_iff {l : Filter α} :
    IntegrableAtFilter f (l ⊓ ae μ) μ ↔ IntegrableAtFilter f l μ 
goal: MeasureTheory.IntegrableAtFilter f (l ⊓ MeasureTheory.ae μ) μ ↔ MeasureTheory.IntegrableAtFilter f l μ

full_name: MeasureTheory.integrableAtFilter_top
code: @[simp]
theorem integrableAtFilter_top : IntegrableAtFilter f ⊤ μ ↔ Integrable f μ 
goal: MeasureTheory.IntegrableAtFilter f ⊤ μ ↔ MeasureTheory.Integrable f μ

full_name: MeasureTheory.IntegrableAtFilter.sup_iff
code: theorem IntegrableAtFilter.sup_iff {l l' : Filter α} :
    IntegrableAtFilter f (l ⊔ l') μ ↔ IntegrableAtFilter f l μ ∧ IntegrableAtFilter f l' μ 
goal: MeasureTheory.IntegrableAtFilter f (l ⊔ l') μ ↔
  MeasureTheory.IntegrableAtFilter f l μ ∧ MeasureTheory.IntegrableAtFilter f l' μ

full_name: MeasureTheory.Measure.FiniteAtFilter.integrableAtFilter
code: /-- If `μ` is a measure finite at filter `l` and `f` is a function such that its norm is bounded
above at `l`, then `f` is integrable at `l`. -/
theorem Measure.FiniteAtFilter.integrableAtFilter {l : Filter α} [IsMeasurablyGenerated l]
    (hfm : StronglyMeasurableAtFilter f l μ) (hμ : μ.FiniteAtFilter l)
    (hf : l.IsBoundedUnder (· ≤ ·) (norm ∘ f)) : IntegrableAtFilter f l μ 
goal: MeasureTheory.IntegrableAtFilter f l μ

full_name: MeasureTheory.Measure.FiniteAtFilter.integrableAtFilter_of_tendsto_ae
code: theorem Measure.FiniteAtFilter.integrableAtFilter_of_tendsto_ae {l : Filter α}
    [IsMeasurablyGenerated l] (hfm : StronglyMeasurableAtFilter f l μ) (hμ : μ.FiniteAtFilter l) {b}
    (hf : Tendsto f (l ⊓ ae μ) (𝓝 b)) : IntegrableAtFilter f l μ 
goal: MeasureTheory.IntegrableAtFilter f l μ

full_name: MeasureTheory.Measure.FiniteAtFilter.integrableAtFilter_of_tendsto
code: theorem Measure.FiniteAtFilter.integrableAtFilter_of_tendsto {l : Filter α}
    [IsMeasurablyGenerated l] (hfm : StronglyMeasurableAtFilter f l μ) (hμ : μ.FiniteAtFilter l) {b}
    (hf : Tendsto f l (𝓝 b)) : IntegrableAtFilter f l μ 
goal: MeasureTheory.IntegrableAtFilter f l μ

full_name: MeasureTheory.Measure.integrableOn_of_bounded
code: lemma Measure.integrableOn_of_bounded (s_finite : μ s ≠ ∞) (f_mble : AEStronglyMeasurable f μ)
    {M : ℝ} (f_bdd : ∀ᵐ a ∂(μ.restrict s), ‖f a‖ ≤ M) :
    IntegrableOn f s μ 
goal: MeasureTheory.IntegrableOn f s μ

full_name: MeasureTheory.integrable_add_of_disjoint
code: theorem integrable_add_of_disjoint {f g : α → E} (h : Disjoint (support f) (support g))
    (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) :
    Integrable (f + g) μ ↔ Integrable f μ ∧ Integrable g μ 
goal: MeasureTheory.Integrable (f + g) μ ↔ MeasureTheory.Integrable f μ ∧ MeasureTheory.Integrable g μ

full_name: MeasureTheory.IntegrableAtFilter.eq_zero_of_tendsto
code: /-- If a function converges along a filter to a limit `a`, is integrable along this filter, and
all elements of the filter have infinite measure, then the limit has to vanish. -/
lemma IntegrableAtFilter.eq_zero_of_tendsto
    (h : IntegrableAtFilter f l μ) (h' : ∀ s ∈ l, μ s = ∞) {a : E}
    (hf : Tendsto f l (𝓝 a)) : a = 0 
goal: a = 0

full_name: ContinuousOn.aemeasurable
code: /-- A function which is continuous on a set `s` is almost everywhere measurable with respect to
`μ.restrict s`. -/
theorem ContinuousOn.aemeasurable [TopologicalSpace α] [OpensMeasurableSpace α] [MeasurableSpace β]
    [TopologicalSpace β] [BorelSpace β] {f : α → β} {s : Set α} {μ : Measure α}
    (hf : ContinuousOn f s) (hs : MeasurableSet s) : AEMeasurable f (μ.restrict s) 
goal: AEMeasurable f (μ.restrict s)

full_name: ContinuousOn.aestronglyMeasurable_of_isSeparable
code: /-- A function which is continuous on a separable set `s` is almost everywhere strongly measurable
with respect to `μ.restrict s`. -/
theorem ContinuousOn.aestronglyMeasurable_of_isSeparable [TopologicalSpace α]
    [PseudoMetrizableSpace α] [OpensMeasurableSpace α] [TopologicalSpace β]
    [PseudoMetrizableSpace β] {f : α → β} {s : Set α} {μ : Measure α} (hf : ContinuousOn f s)
    (hs : MeasurableSet s) (h's : TopologicalSpace.IsSeparable s) :
    AEStronglyMeasurable f (μ.restrict s) 
goal: MeasureTheory.AEStronglyMeasurable f (μ.restrict s)

full_name: ContinuousOn.aestronglyMeasurable
code: /-- A function which is continuous on a set `s` is almost everywhere strongly measurable with
respect to `μ.restrict s` when either the source space or the target space is second-countable. -/
theorem ContinuousOn.aestronglyMeasurable [TopologicalSpace α] [TopologicalSpace β]
    [h : SecondCountableTopologyEither α β] [OpensMeasurableSpace α] [PseudoMetrizableSpace β]
    {f : α → β} {s : Set α} {μ : Measure α} (hf : ContinuousOn f s) (hs : MeasurableSet s) :
    AEStronglyMeasurable f (μ.restrict s) 
goal: MeasureTheory.AEStronglyMeasurable f (μ.restrict s)

full_name: ContinuousOn.aestronglyMeasurable_of_isCompact
code: /-- A function which is continuous on a compact set `s` is almost everywhere strongly measurable
with respect to `μ.restrict s`. -/
theorem ContinuousOn.aestronglyMeasurable_of_isCompact [TopologicalSpace α] [OpensMeasurableSpace α]
    [TopologicalSpace β] [PseudoMetrizableSpace β] {f : α → β} {s : Set α} {μ : Measure α}
    (hf : ContinuousOn f s) (hs : IsCompact s) (h's : MeasurableSet s) :
    AEStronglyMeasurable f (μ.restrict s) 
goal: MeasureTheory.AEStronglyMeasurable f (μ.restrict s)

full_name: ContinuousOn.integrableAt_nhdsWithin_of_isSeparable
code: theorem ContinuousOn.integrableAt_nhdsWithin_of_isSeparable [TopologicalSpace α]
    [PseudoMetrizableSpace α] [OpensMeasurableSpace α] {μ : Measure α} [IsLocallyFiniteMeasure μ]
    {a : α} {t : Set α} {f : α → E} (hft : ContinuousOn f t) (ht : MeasurableSet t)
    (h't : TopologicalSpace.IsSeparable t) (ha : a ∈ t) : IntegrableAtFilter f (𝓝[t] a) μ 
goal: MeasureTheory.IntegrableAtFilter f (𝓝[t] a) μ

full_name: ContinuousOn.integrableAt_nhdsWithin
code: theorem ContinuousOn.integrableAt_nhdsWithin [TopologicalSpace α]
    [SecondCountableTopologyEither α E] [OpensMeasurableSpace α] {μ : Measure α}
    [IsLocallyFiniteMeasure μ] {a : α} {t : Set α} {f : α → E} (hft : ContinuousOn f t)
    (ht : MeasurableSet t) (ha : a ∈ t) : IntegrableAtFilter f (𝓝[t] a) μ 
goal: MeasureTheory.IntegrableAtFilter f (𝓝[t] a) μ

full_name: Continuous.integrableAt_nhds
code: theorem Continuous.integrableAt_nhds [TopologicalSpace α] [SecondCountableTopologyEither α E]
    [OpensMeasurableSpace α] {μ : Measure α} [IsLocallyFiniteMeasure μ] {f : α → E}
    (hf : Continuous f) (a : α) : IntegrableAtFilter f (𝓝 a) μ 
goal: MeasureTheory.IntegrableAtFilter f (𝓝 a) μ

full_name: ContinuousOn.stronglyMeasurableAtFilter
code: /-- If a function is continuous on an open set `s`, then it is strongly measurable at the filter
`𝓝 x` for all `x ∈ s` if either the source space or the target space is second-countable. -/
theorem ContinuousOn.stronglyMeasurableAtFilter [TopologicalSpace α] [OpensMeasurableSpace α]
    [TopologicalSpace β] [PseudoMetrizableSpace β] [SecondCountableTopologyEither α β] {f : α → β}
    {s : Set α} {μ : Measure α} (hs : IsOpen s) (hf : ContinuousOn f s) :
    ∀ x ∈ s, StronglyMeasurableAtFilter f (𝓝 x) μ 
goal: x ∈ s → StronglyMeasurableAtFilter f (𝓝 x) μ

full_name: ContinuousAt.stronglyMeasurableAtFilter
code: theorem ContinuousAt.stronglyMeasurableAtFilter [TopologicalSpace α] [OpensMeasurableSpace α]
    [SecondCountableTopologyEither α E] {f : α → E} {s : Set α} {μ : Measure α} (hs : IsOpen s)
    (hf : ∀ x ∈ s, ContinuousAt f x) : ∀ x ∈ s, StronglyMeasurableAtFilter f (𝓝 x) μ 
goal: x ∈ s → StronglyMeasurableAtFilter f (𝓝 x) μ

full_name: Continuous.stronglyMeasurableAtFilter
code: theorem Continuous.stronglyMeasurableAtFilter [TopologicalSpace α] [OpensMeasurableSpace α]
    [TopologicalSpace β] [PseudoMetrizableSpace β] [SecondCountableTopologyEither α β] {f : α → β}
    (hf : Continuous f) (μ : Measure α) (l : Filter α) : StronglyMeasurableAtFilter f l μ 
goal: StronglyMeasurableAtFilter f l μ

full_name: ContinuousOn.stronglyMeasurableAtFilter_nhdsWithin
code: /-- If a function is continuous on a measurable set `s`, then it is measurable at the filter
  `𝓝[s] x` for all `x`. -/
theorem ContinuousOn.stronglyMeasurableAtFilter_nhdsWithin {α β : Type*} [MeasurableSpace α]
    [TopologicalSpace α] [OpensMeasurableSpace α] [TopologicalSpace β] [PseudoMetrizableSpace β]
    [SecondCountableTopologyEither α β] {f : α → β} {s : Set α} {μ : Measure α}
    (hf : ContinuousOn f s) (hs : MeasurableSet s) (x : α) :
    StronglyMeasurableAtFilter f (𝓝[s] x) μ 
goal: StronglyMeasurableAtFilter f (𝓝[s] x) μ

full_name: integrableOn_Icc_iff_integrableOn_Ioc'
code: theorem integrableOn_Icc_iff_integrableOn_Ioc' (ha : μ {a} ≠ ∞) :
    IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ioc a b) μ 
goal: MeasureTheory.IntegrableOn f (Set.Icc a b) μ ↔ MeasureTheory.IntegrableOn f (Set.Ioc a b) μ

full_name: integrableOn_Icc_iff_integrableOn_Ico'
code: theorem integrableOn_Icc_iff_integrableOn_Ico' (hb : μ {b} ≠ ∞) :
    IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ico a b) μ 
goal: MeasureTheory.IntegrableOn f (Set.Icc a b) μ ↔ MeasureTheory.IntegrableOn f (Set.Ico a b) μ

full_name: integrableOn_Ico_iff_integrableOn_Ioo'
code: theorem integrableOn_Ico_iff_integrableOn_Ioo' (ha : μ {a} ≠ ∞) :
    IntegrableOn f (Ico a b) μ ↔ IntegrableOn f (Ioo a b) μ 
goal: MeasureTheory.IntegrableOn f (Set.Ico a b) μ ↔ MeasureTheory.IntegrableOn f (Set.Ioo a b) μ

full_name: integrableOn_Ioc_iff_integrableOn_Ioo'
code: theorem integrableOn_Ioc_iff_integrableOn_Ioo' (hb : μ {b} ≠ ∞) :
    IntegrableOn f (Ioc a b) μ ↔ IntegrableOn f (Ioo a b) μ 
goal: MeasureTheory.IntegrableOn f (Set.Ioc a b) μ ↔ MeasureTheory.IntegrableOn f (Set.Ioo a b) μ

full_name: integrableOn_Icc_iff_integrableOn_Ioo'
code: theorem integrableOn_Icc_iff_integrableOn_Ioo' (ha : μ {a} ≠ ∞) (hb : μ {b} ≠ ∞) :
    IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ioo a b) μ 
goal: MeasureTheory.IntegrableOn f (Set.Icc a b) μ ↔ MeasureTheory.IntegrableOn f (Set.Ioo a b) μ

full_name: integrableOn_Ici_iff_integrableOn_Ioi'
code: theorem integrableOn_Ici_iff_integrableOn_Ioi' (hb : μ {b} ≠ ∞) :
    IntegrableOn f (Ici b) μ ↔ IntegrableOn f (Ioi b) μ 
goal: MeasureTheory.IntegrableOn f (Set.Ici b) μ ↔ MeasureTheory.IntegrableOn f (Set.Ioi b) μ

full_name: integrableOn_Iic_iff_integrableOn_Iio'
code: theorem integrableOn_Iic_iff_integrableOn_Iio' (hb : μ {b} ≠ ∞) :
    IntegrableOn f (Iic b) μ ↔ IntegrableOn f (Iio b) μ 
goal: MeasureTheory.IntegrableOn f (Set.Iic b) μ ↔ MeasureTheory.IntegrableOn f (Set.Iio b) μ

full_name: integrableOn_Icc_iff_integrableOn_Ioc
code: theorem integrableOn_Icc_iff_integrableOn_Ioc :
    IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ioc a b) μ 
goal: MeasureTheory.IntegrableOn f (Set.Icc a b) μ ↔ MeasureTheory.IntegrableOn f (Set.Ioc a b) μ

full_name: integrableOn_Icc_iff_integrableOn_Ico
code: theorem integrableOn_Icc_iff_integrableOn_Ico :
    IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ico a b) μ 
goal: MeasureTheory.IntegrableOn f (Set.Icc a b) μ ↔ MeasureTheory.IntegrableOn f (Set.Ico a b) μ

full_name: integrableOn_Ico_iff_integrableOn_Ioo
code: theorem integrableOn_Ico_iff_integrableOn_Ioo :
    IntegrableOn f (Ico a b) μ ↔ IntegrableOn f (Ioo a b) μ 
goal: MeasureTheory.IntegrableOn f (Set.Ico a b) μ ↔ MeasureTheory.IntegrableOn f (Set.Ioo a b) μ

full_name: integrableOn_Ioc_iff_integrableOn_Ioo
code: theorem integrableOn_Ioc_iff_integrableOn_Ioo :
    IntegrableOn f (Ioc a b) μ ↔ IntegrableOn f (Ioo a b) μ 
goal: MeasureTheory.IntegrableOn f (Set.Ioc a b) μ ↔ MeasureTheory.IntegrableOn f (Set.Ioo a b) μ

full_name: integrableOn_Icc_iff_integrableOn_Ioo
code: theorem integrableOn_Icc_iff_integrableOn_Ioo :
    IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ioo a b) μ 
goal: MeasureTheory.IntegrableOn f (Set.Icc a b) μ ↔ MeasureTheory.IntegrableOn f (Set.Ioo a b) μ

full_name: integrableOn_Ici_iff_integrableOn_Ioi
code: theorem integrableOn_Ici_iff_integrableOn_Ioi :
    IntegrableOn f (Ici b) μ ↔ IntegrableOn f (Ioi b) μ 
goal: MeasureTheory.IntegrableOn f (Set.Ici b) μ ↔ MeasureTheory.IntegrableOn f (Set.Ioi b) μ

full_name: integrableOn_Iic_iff_integrableOn_Iio
code: theorem integrableOn_Iic_iff_integrableOn_Iio :
    IntegrableOn f (Iic b) μ ↔ IntegrableOn f (Iio b) μ 
goal: MeasureTheory.IntegrableOn f (Set.Iic b) μ ↔ MeasureTheory.IntegrableOn f (Set.Iio b) μ