File: Mathlib/MeasureTheory/Function/LocallyIntegrable.lean

full_name: MeasureTheory.LocallyIntegrableOn
code: /-- A function `f : X → E` is *locally integrable on s*, for `s ⊆ X`, if for every `x ∈ s` there is
a neighbourhood of `x` within `s` on which `f` is integrable. (Note this is, in general, strictly
weaker than local integrability with respect to `μ.restrict s`.) -/
def LocallyIntegrableOn (f : X → E) (s : Set X) (μ : Measure X := by volume_tac) : Prop 
goal: Prop

full_name: MeasureTheory.LocallyIntegrableOn.mono_set
code: theorem LocallyIntegrableOn.mono_set (hf : LocallyIntegrableOn f s μ) {t : Set X}
    (hst : t ⊆ s) : LocallyIntegrableOn f t μ 
goal: MeasureTheory.LocallyIntegrableOn f t μ

full_name: MeasureTheory.LocallyIntegrableOn.norm
code: theorem LocallyIntegrableOn.norm (hf : LocallyIntegrableOn f s μ) :
    LocallyIntegrableOn (fun x => ‖f x‖) s μ 
goal: MeasureTheory.LocallyIntegrableOn (fun x => ‖f x‖) s μ

full_name: MeasureTheory.LocallyIntegrableOn.mono
code: theorem LocallyIntegrableOn.mono (hf : LocallyIntegrableOn f s μ) {g : X → F}
    (hg : AEStronglyMeasurable g μ) (h : ∀ᵐ x ∂μ, ‖g x‖ ≤ ‖f x‖) :
    LocallyIntegrableOn g s μ 
goal: MeasureTheory.LocallyIntegrableOn g s μ

full_name: MeasureTheory.IntegrableOn.locallyIntegrableOn
code: theorem IntegrableOn.locallyIntegrableOn (hf : IntegrableOn f s μ) : LocallyIntegrableOn f s μ 
goal: MeasureTheory.LocallyIntegrableOn f s μ

full_name: MeasureTheory.LocallyIntegrableOn.integrableOn_isCompact
code: /-- If a function is locally integrable on a compact set, then it is integrable on that set. -/
theorem LocallyIntegrableOn.integrableOn_isCompact (hf : LocallyIntegrableOn f s μ)
    (hs : IsCompact s) : IntegrableOn f s μ 
goal: MeasureTheory.IntegrableOn f s μ

full_name: MeasureTheory.LocallyIntegrableOn.integrableOn_compact_subset
code: theorem LocallyIntegrableOn.integrableOn_compact_subset (hf : LocallyIntegrableOn f s μ) {t : Set X}
    (hst : t ⊆ s) (ht : IsCompact t) : IntegrableOn f t μ 
goal: MeasureTheory.IntegrableOn f t μ

full_name: MeasureTheory.LocallyIntegrableOn.exists_countable_integrableOn
code: /-- If a function `f` is locally integrable on a set `s` in a second countable topological space,
then there exist countably many open sets `u` covering `s` such that `f` is integrable on each
set `u ∩ s`. -/
theorem LocallyIntegrableOn.exists_countable_integrableOn [SecondCountableTopology X]
    (hf : LocallyIntegrableOn f s μ) : ∃ T : Set (Set X), T.Countable ∧
    (∀ u ∈ T, IsOpen u) ∧ (s ⊆ ⋃ u ∈ T, u) ∧ (∀ u ∈ T, IntegrableOn f (u ∩ s) μ) 
goal: ∃ T, T.Countable ∧ (∀ u ∈ T, IsOpen u) ∧ s ⊆ ⋃ u ∈ T, u ∧ ∀ u ∈ T, MeasureTheory.IntegrableOn f (u ∩ s) μ

full_name: MeasureTheory.LocallyIntegrableOn.exists_nat_integrableOn
code: /-- If a function `f` is locally integrable on a set `s` in a second countable topological space,
then there exists a sequence of open sets `u n` covering `s` such that `f` is integrable on each
set `u n ∩ s`. -/
theorem LocallyIntegrableOn.exists_nat_integrableOn [SecondCountableTopology X]
    (hf : LocallyIntegrableOn f s μ) : ∃ u : ℕ → Set X,
    (∀ n, IsOpen (u n)) ∧ (s ⊆ ⋃ n, u n) ∧ (∀ n, IntegrableOn f (u n ∩ s) μ) 
goal: ∃ u, (∀ (n : ℕ), IsOpen (u n)) ∧ s ⊆ ⋃ n, u n ∧ ∀ (n : ℕ), MeasureTheory.IntegrableOn f (u n ∩ s) μ

full_name: MeasureTheory.LocallyIntegrableOn.aestronglyMeasurable
code: theorem LocallyIntegrableOn.aestronglyMeasurable [SecondCountableTopology X]
    (hf : LocallyIntegrableOn f s μ) : AEStronglyMeasurable f (μ.restrict s) 
goal: MeasureTheory.AEStronglyMeasurable f (μ.restrict s)

full_name: MeasureTheory.locallyIntegrableOn_iff
code: /-- If `s` is either open, or closed, then `f` is locally integrable on `s` iff it is integrable on
every compact subset contained in `s`. -/
theorem locallyIntegrableOn_iff [LocallyCompactSpace X] [T2Space X] (hs : IsClosed s ∨ IsOpen s) :
    LocallyIntegrableOn f s μ ↔ ∀ (k : Set X), k ⊆ s → (IsCompact k → IntegrableOn f k μ) 
goal: MeasureTheory.LocallyIntegrableOn f s μ ↔ ∀ k ⊆ s, IsCompact k → MeasureTheory.IntegrableOn f k μ

full_name: MeasureTheory.LocallyIntegrableOn.add
code: protected theorem LocallyIntegrableOn.add
    (hf : LocallyIntegrableOn f s μ) (hg : LocallyIntegrableOn g s μ) :
    LocallyIntegrableOn (f + g) s μ 
goal: MeasureTheory.LocallyIntegrableOn (f + g) s μ

full_name: MeasureTheory.LocallyIntegrableOn.sub
code: protected theorem LocallyIntegrableOn.sub
    (hf : LocallyIntegrableOn f s μ) (hg : LocallyIntegrableOn g s μ) :
    LocallyIntegrableOn (f - g) s μ 
goal: MeasureTheory.LocallyIntegrableOn (f - g) s μ

full_name: MeasureTheory.LocallyIntegrableOn.neg
code: protected theorem LocallyIntegrableOn.neg (hf : LocallyIntegrableOn f s μ) :
    LocallyIntegrableOn (-f) s μ 
goal: MeasureTheory.LocallyIntegrableOn (-f) s μ

full_name: MeasureTheory.LocallyIntegrable
code: /-- A function `f : X → E` is *locally integrable* if it is integrable on a neighborhood of every
point. In particular, it is integrable on all compact sets,
see `LocallyIntegrable.integrableOn_isCompact`. -/
def LocallyIntegrable (f : X → E) (μ : Measure X := by volume_tac) : Prop 
goal: Prop

full_name: MeasureTheory.locallyIntegrable_comap
code: theorem locallyIntegrable_comap (hs : MeasurableSet s) :
    LocallyIntegrable (fun x : s ↦ f x) (μ.comap Subtype.val) ↔ LocallyIntegrableOn f s μ 
goal: MeasureTheory.LocallyIntegrable (fun x => f ↑x) (MeasureTheory.Measure.comap Subtype.val μ) ↔
  MeasureTheory.LocallyIntegrableOn f s μ

full_name: MeasureTheory.locallyIntegrableOn_univ
code: theorem locallyIntegrableOn_univ : LocallyIntegrableOn f univ μ ↔ LocallyIntegrable f μ 
goal: MeasureTheory.LocallyIntegrableOn f Set.univ μ ↔ MeasureTheory.LocallyIntegrable f μ

full_name: MeasureTheory.LocallyIntegrable.locallyIntegrableOn
code: theorem LocallyIntegrable.locallyIntegrableOn (hf : LocallyIntegrable f μ) (s : Set X) :
    LocallyIntegrableOn f s μ 
goal: MeasureTheory.LocallyIntegrableOn f s μ

full_name: MeasureTheory.Integrable.locallyIntegrable
code: theorem Integrable.locallyIntegrable (hf : Integrable f μ) : LocallyIntegrable f μ 
goal: MeasureTheory.LocallyIntegrable f μ

full_name: MeasureTheory.LocallyIntegrable.mono
code: theorem LocallyIntegrable.mono (hf : LocallyIntegrable f μ) {g : X → F}
    (hg : AEStronglyMeasurable g μ) (h : ∀ᵐ x ∂μ, ‖g x‖ ≤ ‖f x‖) :
    LocallyIntegrable g μ 
goal: MeasureTheory.LocallyIntegrable g μ

full_name: MeasureTheory.locallyIntegrableOn_of_locallyIntegrable_restrict
code: /-- If `f` is locally integrable with respect to `μ.restrict s`, it is locally integrable on `s`.
(See `locallyIntegrableOn_iff_locallyIntegrable_restrict` for an iff statement when `s` is
closed.) -/
theorem locallyIntegrableOn_of_locallyIntegrable_restrict [OpensMeasurableSpace X]
    (hf : LocallyIntegrable f (μ.restrict s)) : LocallyIntegrableOn f s μ 
goal: MeasureTheory.LocallyIntegrableOn f s μ

full_name: MeasureTheory.locallyIntegrableOn_iff_locallyIntegrable_restrict
code: /-- If `s` is closed, being locally integrable on `s` wrt `μ` is equivalent to being locally
integrable with respect to `μ.restrict s`. For the one-way implication without assuming `s` closed,
see `locallyIntegrableOn_of_locallyIntegrable_restrict`. -/
theorem locallyIntegrableOn_iff_locallyIntegrable_restrict [OpensMeasurableSpace X]
    (hs : IsClosed s) : LocallyIntegrableOn f s μ ↔ LocallyIntegrable f (μ.restrict s) 
goal: MeasureTheory.LocallyIntegrableOn f s μ ↔ MeasureTheory.LocallyIntegrable f (μ.restrict s)

full_name: MeasureTheory.LocallyIntegrable.integrableOn_isCompact
code: /-- If a function is locally integrable, then it is integrable on any compact set. -/
theorem LocallyIntegrable.integrableOn_isCompact {k : Set X} (hf : LocallyIntegrable f μ)
    (hk : IsCompact k) : IntegrableOn f k μ 
goal: MeasureTheory.IntegrableOn f k μ

full_name: MeasureTheory.LocallyIntegrable.integrableOn_nhds_isCompact
code: /-- If a function is locally integrable, then it is integrable on an open neighborhood of any
compact set. -/
theorem LocallyIntegrable.integrableOn_nhds_isCompact (hf : LocallyIntegrable f μ) {k : Set X}
    (hk : IsCompact k) : ∃ u, IsOpen u ∧ k ⊆ u ∧ IntegrableOn f u μ 
goal: ∃ u, IsOpen u ∧ k ⊆ u ∧ MeasureTheory.IntegrableOn f u μ

full_name: MeasureTheory.locallyIntegrable_iff
code: theorem locallyIntegrable_iff [LocallyCompactSpace X] :
    LocallyIntegrable f μ ↔ ∀ k : Set X, IsCompact k → IntegrableOn f k μ 
goal: MeasureTheory.LocallyIntegrable f μ ↔ ∀ (k : Set X), IsCompact k → MeasureTheory.IntegrableOn f k μ

full_name: MeasureTheory.LocallyIntegrable.aestronglyMeasurable
code: theorem LocallyIntegrable.aestronglyMeasurable [SecondCountableTopology X]
    (hf : LocallyIntegrable f μ) : AEStronglyMeasurable f μ 
goal: MeasureTheory.AEStronglyMeasurable f μ

full_name: MeasureTheory.LocallyIntegrable.exists_nat_integrableOn
code: /-- If a function is locally integrable in a second countable topological space,
then there exists a sequence of open sets covering the space on which it is integrable. -/
theorem LocallyIntegrable.exists_nat_integrableOn [SecondCountableTopology X]
    (hf : LocallyIntegrable f μ) : ∃ u : ℕ → Set X,
    (∀ n, IsOpen (u n)) ∧ ((⋃ n, u n) = univ) ∧ (∀ n, IntegrableOn f (u n) μ) 
goal: ∃ u, (∀ (n : ℕ), IsOpen (u n)) ∧ ⋃ n, u n = Set.univ ∧ ∀ (n : ℕ), MeasureTheory.IntegrableOn f (u n) μ

full_name: MeasureTheory.Memℒp.locallyIntegrable
code: theorem Memℒp.locallyIntegrable [IsLocallyFiniteMeasure μ] {f : X → E} {p : ℝ≥0∞}
    (hf : Memℒp f p μ) (hp : 1 ≤ p) : LocallyIntegrable f μ 
goal: MeasureTheory.LocallyIntegrable f μ

full_name: MeasureTheory.locallyIntegrable_const
code: theorem locallyIntegrable_const [IsLocallyFiniteMeasure μ] (c : E) :
    LocallyIntegrable (fun _ => c) μ 
goal: MeasureTheory.LocallyIntegrable (fun x => c) μ

full_name: MeasureTheory.locallyIntegrableOn_const
code: theorem locallyIntegrableOn_const [IsLocallyFiniteMeasure μ] (c : E) :
    LocallyIntegrableOn (fun _ => c) s μ 
goal: MeasureTheory.LocallyIntegrableOn (fun x => c) s μ

full_name: MeasureTheory.locallyIntegrable_zero
code: theorem locallyIntegrable_zero : LocallyIntegrable (fun _ ↦ (0 : E)) μ 
goal: MeasureTheory.LocallyIntegrable (fun x => 0) μ

full_name: MeasureTheory.locallyIntegrableOn_zero
code: theorem locallyIntegrableOn_zero : LocallyIntegrableOn (fun _ ↦ (0 : E)) s μ 
goal: MeasureTheory.LocallyIntegrableOn (fun x => 0) s μ

full_name: MeasureTheory.LocallyIntegrable.indicator
code: theorem LocallyIntegrable.indicator (hf : LocallyIntegrable f μ) {s : Set X}
    (hs : MeasurableSet s) : LocallyIntegrable (s.indicator f) μ 
goal: MeasureTheory.LocallyIntegrable (s.indicator f) μ

full_name: MeasureTheory.locallyIntegrable_map_homeomorph
code: theorem locallyIntegrable_map_homeomorph [BorelSpace X] [BorelSpace Y] (e : X ≃ₜ Y) {f : Y → E}
    {μ : Measure X} : LocallyIntegrable f (Measure.map e μ) ↔ LocallyIntegrable (f ∘ e) μ 
goal: MeasureTheory.LocallyIntegrable f (MeasureTheory.Measure.map (⇑e) μ) ↔ MeasureTheory.LocallyIntegrable (f ∘ ⇑e) μ

full_name: MeasureTheory.LocallyIntegrable.add
code: protected theorem LocallyIntegrable.add (hf : LocallyIntegrable f μ) (hg : LocallyIntegrable g μ) :
    LocallyIntegrable (f + g) μ 
goal: MeasureTheory.LocallyIntegrable (f + g) μ

full_name: MeasureTheory.LocallyIntegrable.sub
code: protected theorem LocallyIntegrable.sub (hf : LocallyIntegrable f μ) (hg : LocallyIntegrable g μ) :
    LocallyIntegrable (f - g) μ 
goal: MeasureTheory.LocallyIntegrable (f - g) μ

full_name: MeasureTheory.LocallyIntegrable.neg
code: protected theorem LocallyIntegrable.neg (hf : LocallyIntegrable f μ) :
    LocallyIntegrable (-f) μ 
goal: MeasureTheory.LocallyIntegrable (-f) μ

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

full_name: MeasureTheory.locallyIntegrable_finset_sum'
code: theorem locallyIntegrable_finset_sum' {ι} (s : Finset ι) {f : ι → X → E}
    (hf : ∀ i ∈ s, LocallyIntegrable (f i) μ) : LocallyIntegrable (∑ i ∈ s, f i) μ 
goal: MeasureTheory.LocallyIntegrable (∑ i ∈ s, f i) μ

full_name: MeasureTheory.locallyIntegrable_finset_sum
code: theorem locallyIntegrable_finset_sum {ι} (s : Finset ι) {f : ι → X → E}
    (hf : ∀ i ∈ s, LocallyIntegrable (f i) μ) : LocallyIntegrable (fun a ↦ ∑ i ∈ s, f i a) μ 
goal: MeasureTheory.LocallyIntegrable (fun a => ∑ i ∈ s, f i a) μ

full_name: MeasureTheory.LocallyIntegrable.integrable_smul_left_of_hasCompactSupport
code: /-- If `f` is locally integrable and `g` is continuous with compact support,
then `g • f` is integrable. -/
theorem LocallyIntegrable.integrable_smul_left_of_hasCompactSupport
    [NormedSpace ℝ E] [OpensMeasurableSpace X] [T2Space X]
    (hf : LocallyIntegrable f μ) {g : X → ℝ} (hg : Continuous g) (h'g : HasCompactSupport g) :
    Integrable (fun x ↦ g x • f x) μ 
goal: MeasureTheory.Integrable (fun x => g x • f x) μ

full_name: MeasureTheory.LocallyIntegrable.integrable_smul_right_of_hasCompactSupport
code: /-- If `f` is locally integrable and `g` is continuous with compact support,
then `f • g` is integrable. -/
theorem LocallyIntegrable.integrable_smul_right_of_hasCompactSupport
    [NormedSpace ℝ E] [OpensMeasurableSpace X] [T2Space X] {f : X → ℝ} (hf : LocallyIntegrable f μ)
    {g : X → E} (hg : Continuous g) (h'g : HasCompactSupport g) :
    Integrable (fun x ↦ f x • g x) μ 
goal: MeasureTheory.Integrable (fun x => f x • g x) μ

full_name: MeasureTheory.integrable_iff_integrableAtFilter_cocompact
code: theorem integrable_iff_integrableAtFilter_cocompact :
    Integrable f μ ↔ (IntegrableAtFilter f (cocompact X) μ ∧ LocallyIntegrable f μ) 
goal: MeasureTheory.Integrable f μ ↔
  MeasureTheory.IntegrableAtFilter f (Filter.cocompact X) μ ∧ MeasureTheory.LocallyIntegrable f μ

full_name: MeasureTheory.integrable_iff_integrableAtFilter_atBot_atTop
code: theorem integrable_iff_integrableAtFilter_atBot_atTop [LinearOrder X] [CompactIccSpace X] :
    Integrable f μ ↔
    (IntegrableAtFilter f atBot μ ∧ IntegrableAtFilter f atTop μ) ∧ LocallyIntegrable f μ 
goal: MeasureTheory.Integrable f μ ↔
  (MeasureTheory.IntegrableAtFilter f Filter.atBot μ ∧ MeasureTheory.IntegrableAtFilter f Filter.atTop μ) ∧
    MeasureTheory.LocallyIntegrable f μ

full_name: MeasureTheory.integrable_iff_integrableAtFilter_atBot
code: theorem integrable_iff_integrableAtFilter_atBot [LinearOrder X] [OrderTop X] [CompactIccSpace X] :
    Integrable f μ ↔ IntegrableAtFilter f atBot μ ∧ LocallyIntegrable f μ 
goal: MeasureTheory.Integrable f μ ↔ MeasureTheory.IntegrableAtFilter f Filter.atBot μ ∧ MeasureTheory.LocallyIntegrable f μ

full_name: MeasureTheory.integrable_iff_integrableAtFilter_atTop
code: theorem integrable_iff_integrableAtFilter_atTop [LinearOrder X] [OrderBot X] [CompactIccSpace X] :
    Integrable f μ ↔ IntegrableAtFilter f atTop μ ∧ LocallyIntegrable f μ 
goal: MeasureTheory.Integrable f μ ↔ MeasureTheory.IntegrableAtFilter f Filter.atTop μ ∧ MeasureTheory.LocallyIntegrable f μ

full_name: MeasureTheory.integrableOn_Iic_iff_integrableAtFilter_atBot
code: theorem integrableOn_Iic_iff_integrableAtFilter_atBot [LinearOrder X] [CompactIccSpace X] :
    IntegrableOn f (Iic a) μ ↔ IntegrableAtFilter f atBot μ ∧ LocallyIntegrableOn f (Iic a) μ 
goal: MeasureTheory.IntegrableOn f (Set.Iic a) μ ↔
  MeasureTheory.IntegrableAtFilter f Filter.atBot μ ∧ MeasureTheory.LocallyIntegrableOn f (Set.Iic a) μ

full_name: MeasureTheory.integrableOn_Ici_iff_integrableAtFilter_atTop
code: theorem integrableOn_Ici_iff_integrableAtFilter_atTop [LinearOrder X] [CompactIccSpace X] :
    IntegrableOn f (Ici a) μ ↔ IntegrableAtFilter f atTop μ ∧ LocallyIntegrableOn f (Ici a) μ 
goal: MeasureTheory.IntegrableOn f (Set.Ici a) μ ↔
  MeasureTheory.IntegrableAtFilter f Filter.atTop μ ∧ MeasureTheory.LocallyIntegrableOn f (Set.Ici a) μ

full_name: MeasureTheory.integrableOn_Iio_iff_integrableAtFilter_atBot_nhdsWithin
code: theorem integrableOn_Iio_iff_integrableAtFilter_atBot_nhdsWithin
    [LinearOrder X] [CompactIccSpace X] [NoMinOrder X] [OrderTopology X] :
    IntegrableOn f (Iio a) μ ↔ IntegrableAtFilter f atBot μ ∧
    IntegrableAtFilter f (𝓝[<] a) μ ∧ LocallyIntegrableOn f (Iio a) μ 
goal: MeasureTheory.IntegrableOn f (Set.Iio a) μ ↔
  MeasureTheory.IntegrableAtFilter f Filter.atBot μ ∧
    MeasureTheory.IntegrableAtFilter f (𝓝[<] a) μ ∧ MeasureTheory.LocallyIntegrableOn f (Set.Iio a) μ

full_name: MeasureTheory.integrableOn_Ioi_iff_integrableAtFilter_atTop_nhdsWithin
code: theorem integrableOn_Ioi_iff_integrableAtFilter_atTop_nhdsWithin
    [LinearOrder X] [CompactIccSpace X] [NoMaxOrder X] [OrderTopology X] :
    IntegrableOn f (Ioi a) μ ↔ IntegrableAtFilter f atTop μ ∧
    IntegrableAtFilter f (𝓝[>] a) μ ∧ LocallyIntegrableOn f (Ioi a) μ 
goal: MeasureTheory.IntegrableOn f (Set.Ioi a) μ ↔
  MeasureTheory.IntegrableAtFilter f Filter.atTop μ ∧
    MeasureTheory.IntegrableAtFilter f (𝓝[>] a) μ ∧ MeasureTheory.LocallyIntegrableOn f (Set.Ioi a) μ

full_name: Continuous.locallyIntegrable
code: /-- A continuous function `f` is locally integrable with respect to any locally finite measure. -/
theorem Continuous.locallyIntegrable [IsLocallyFiniteMeasure μ] [SecondCountableTopologyEither X E]
    (hf : Continuous f) : LocallyIntegrable f μ 
goal: MeasureTheory.LocallyIntegrable f μ

full_name: ContinuousOn.locallyIntegrableOn
code: /-- A function `f` continuous on a set `K` is locally integrable on this set with respect
to any locally finite measure. -/
theorem ContinuousOn.locallyIntegrableOn [IsLocallyFiniteMeasure μ]
    [SecondCountableTopologyEither X E] (hf : ContinuousOn f K)
    (hK : MeasurableSet K) : LocallyIntegrableOn f K μ 
goal: MeasureTheory.LocallyIntegrableOn f K μ

full_name: ContinuousOn.integrableOn_compact'
code: /-- A function `f` continuous on a compact set `K` is integrable on this set with respect to any
locally finite measure. -/
theorem ContinuousOn.integrableOn_compact'
    (hK : IsCompact K) (h'K : MeasurableSet K) (hf : ContinuousOn f K) :
    IntegrableOn f K μ 
goal: MeasureTheory.IntegrableOn f K μ

full_name: ContinuousOn.integrableOn_compact
code: theorem ContinuousOn.integrableOn_compact [T2Space X]
    (hK : IsCompact K) (hf : ContinuousOn f K) : IntegrableOn f K μ 
goal: MeasureTheory.IntegrableOn f K μ

full_name: ContinuousOn.integrableOn_Icc
code: theorem ContinuousOn.integrableOn_Icc [Preorder X] [CompactIccSpace X] [T2Space X]
    (hf : ContinuousOn f (Icc a b)) : IntegrableOn f (Icc a b) μ 
goal: MeasureTheory.IntegrableOn f (Set.Icc a b) μ

full_name: Continuous.integrableOn_Icc
code: theorem Continuous.integrableOn_Icc [Preorder X] [CompactIccSpace X] [T2Space X]
    (hf : Continuous f) : IntegrableOn f (Icc a b) μ 
goal: MeasureTheory.IntegrableOn f (Set.Icc a b) μ

full_name: Continuous.integrableOn_Ioc
code: theorem Continuous.integrableOn_Ioc [Preorder X] [CompactIccSpace X] [T2Space X]
    (hf : Continuous f) : IntegrableOn f (Ioc a b) μ 
goal: MeasureTheory.IntegrableOn f (Set.Ioc a b) μ

full_name: ContinuousOn.integrableOn_uIcc
code: theorem ContinuousOn.integrableOn_uIcc [LinearOrder X] [CompactIccSpace X] [T2Space X]
    (hf : ContinuousOn f [[a, b]]) : IntegrableOn f [[a, b]] μ 
goal: MeasureTheory.IntegrableOn f [[a, b]] μ

full_name: Continuous.integrableOn_uIcc
code: theorem Continuous.integrableOn_uIcc [LinearOrder X] [CompactIccSpace X] [T2Space X]
    (hf : Continuous f) : IntegrableOn f [[a, b]] μ 
goal: MeasureTheory.IntegrableOn f [[a, b]] μ

full_name: Continuous.integrableOn_uIoc
code: theorem Continuous.integrableOn_uIoc [LinearOrder X] [CompactIccSpace X] [T2Space X]
    (hf : Continuous f) : IntegrableOn f (Ι a b) μ 
goal: MeasureTheory.IntegrableOn f (Ι a b) μ

full_name: Continuous.integrable_of_hasCompactSupport
code: /-- A continuous function with compact support is integrable on the whole space. -/
theorem Continuous.integrable_of_hasCompactSupport (hf : Continuous f) (hcf : HasCompactSupport f) :
    Integrable f μ 
goal: MeasureTheory.Integrable f μ

full_name: MonotoneOn.integrableOn_of_measure_ne_top
code: theorem MonotoneOn.integrableOn_of_measure_ne_top (hmono : MonotoneOn f s) {a b : X}
    (ha : IsLeast s a) (hb : IsGreatest s b) (hs : μ s ≠ ∞) (h's : MeasurableSet s) :
    IntegrableOn f s μ 
goal: MeasureTheory.IntegrableOn f s μ

full_name: MonotoneOn.integrableOn_isCompact
code: theorem MonotoneOn.integrableOn_isCompact [IsFiniteMeasureOnCompacts μ] (hs : IsCompact s)
    (hmono : MonotoneOn f s) : IntegrableOn f s μ 
goal: MeasureTheory.IntegrableOn f s μ

full_name: AntitoneOn.integrableOn_of_measure_ne_top
code: theorem AntitoneOn.integrableOn_of_measure_ne_top (hanti : AntitoneOn f s) {a b : X}
    (ha : IsLeast s a) (hb : IsGreatest s b) (hs : μ s ≠ ∞) (h's : MeasurableSet s) :
    IntegrableOn f s μ 
goal: MeasureTheory.IntegrableOn f s μ

full_name: AntioneOn.integrableOn_isCompact
code: theorem AntioneOn.integrableOn_isCompact [IsFiniteMeasureOnCompacts μ] (hs : IsCompact s)
    (hanti : AntitoneOn f s) : IntegrableOn f s μ 
goal: MeasureTheory.IntegrableOn f s μ

full_name: Monotone.locallyIntegrable
code: theorem Monotone.locallyIntegrable [IsLocallyFiniteMeasure μ] (hmono : Monotone f) :
    LocallyIntegrable f μ 
goal: MeasureTheory.LocallyIntegrable f μ

full_name: Antitone.locallyIntegrable
code: theorem Antitone.locallyIntegrable [IsLocallyFiniteMeasure μ] (hanti : Antitone f) :
    LocallyIntegrable f μ 
goal: MeasureTheory.LocallyIntegrable f μ

full_name: MeasureTheory.IntegrableOn.mul_continuousOn_of_subset
code: theorem IntegrableOn.mul_continuousOn_of_subset (hg : IntegrableOn g A μ) (hg' : ContinuousOn g' K)
    (hA : MeasurableSet A) (hK : IsCompact K) (hAK : A ⊆ K) :
    IntegrableOn (fun x => g x * g' x) A μ 
goal: MeasureTheory.IntegrableOn (fun x => g x * g' x) A μ

full_name: MeasureTheory.IntegrableOn.mul_continuousOn
code: theorem IntegrableOn.mul_continuousOn [T2Space X] (hg : IntegrableOn g K μ)
    (hg' : ContinuousOn g' K) (hK : IsCompact K) : IntegrableOn (fun x => g x * g' x) K μ 
goal: MeasureTheory.IntegrableOn (fun x => g x * g' x) K μ

full_name: MeasureTheory.IntegrableOn.continuousOn_mul_of_subset
code: theorem IntegrableOn.continuousOn_mul_of_subset (hg : ContinuousOn g K) (hg' : IntegrableOn g' A μ)
    (hK : IsCompact K) (hA : MeasurableSet A) (hAK : A ⊆ K) :
    IntegrableOn (fun x => g x * g' x) A μ 
goal: MeasureTheory.IntegrableOn (fun x => g x * g' x) A μ

full_name: MeasureTheory.IntegrableOn.continuousOn_mul
code: theorem IntegrableOn.continuousOn_mul [T2Space X] (hg : ContinuousOn g K)
    (hg' : IntegrableOn g' K μ) (hK : IsCompact K) : IntegrableOn (fun x => g x * g' x) K μ 
goal: MeasureTheory.IntegrableOn (fun x => g x * g' x) K μ

full_name: MeasureTheory.IntegrableOn.continuousOn_smul
code: theorem IntegrableOn.continuousOn_smul [T2Space X] [SecondCountableTopologyEither X 𝕜] {g : X → E}
    (hg : IntegrableOn g K μ) {f : X → 𝕜} (hf : ContinuousOn f K) (hK : IsCompact K) :
    IntegrableOn (fun x => f x • g x) K μ 
goal: MeasureTheory.IntegrableOn (fun x => f x • g x) K μ

full_name: MeasureTheory.IntegrableOn.smul_continuousOn
code: theorem IntegrableOn.smul_continuousOn [T2Space X] [SecondCountableTopologyEither X E] {f : X → 𝕜}
    (hf : IntegrableOn f K μ) {g : X → E} (hg : ContinuousOn g K) (hK : IsCompact K) :
    IntegrableOn (fun x => f x • g x) K μ 
goal: MeasureTheory.IntegrableOn (fun x => f x • g x) K μ

full_name: MeasureTheory.LocallyIntegrableOn.continuousOn_mul
code: theorem continuousOn_mul [LocallyCompactSpace X] [T2Space X] [NormedRing R]
    [SecondCountableTopologyEither X R] {f g : X → R} {s : Set X} (hf : LocallyIntegrableOn f s μ)
    (hg : ContinuousOn g s) (hs : IsOpen s) : LocallyIntegrableOn (fun x => g x * f x) s μ 
goal: MeasureTheory.LocallyIntegrableOn (fun x => g x * f x) s μ

full_name: MeasureTheory.LocallyIntegrableOn.mul_continuousOn
code: theorem mul_continuousOn [LocallyCompactSpace X] [T2Space X] [NormedRing R]
    [SecondCountableTopologyEither X R] {f g : X → R} {s : Set X} (hf : LocallyIntegrableOn f s μ)
    (hg : ContinuousOn g s) (hs : IsOpen s) : LocallyIntegrableOn (fun x => f x * g x) s μ 
goal: MeasureTheory.LocallyIntegrableOn (fun x => f x * g x) s μ

full_name: MeasureTheory.LocallyIntegrableOn.continuousOn_smul
code: theorem continuousOn_smul [LocallyCompactSpace X] [T2Space X] {𝕜 : Type*} [NormedField 𝕜]
    [SecondCountableTopologyEither X 𝕜] [NormedSpace 𝕜 E] {f : X → E} {g : X → 𝕜} {s : Set X}
    (hs : IsOpen s) (hf : LocallyIntegrableOn f s μ) (hg : ContinuousOn g s) :
    LocallyIntegrableOn (fun x => g x • f x) s μ 
goal: MeasureTheory.LocallyIntegrableOn (fun x => g x • f x) s μ

full_name: MeasureTheory.LocallyIntegrableOn.smul_continuousOn
code: theorem smul_continuousOn [LocallyCompactSpace X] [T2Space X] {𝕜 : Type*} [NormedField 𝕜]
    [SecondCountableTopologyEither X E] [NormedSpace 𝕜 E] {f : X → 𝕜} {g : X → E} {s : Set X}
    (hs : IsOpen s) (hf : LocallyIntegrableOn f s μ) (hg : ContinuousOn g s) :
    LocallyIntegrableOn (fun x => f x • g x) s μ 
goal: MeasureTheory.LocallyIntegrableOn (fun x => f x • g x) s μ