ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.AEEqFun.lintegral_add
theorem lintegral_add (f g : α →ₘ[μ] ℝ≥0∞) : lintegral (f + g) = lintegral f + lintegral g 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} (f g : α →ₘ[μ] ℝ≥0∞)
Goal: (f + g).lintegral = f.lintegral + g.lintegral
-- MeasureTheory.AEEqFun.lintegral_mono
theorem lintegral_mono {f g : α →ₘ[μ] ℝ≥0∞} : f ≤ g → lintegral f ≤ lintegral g 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {f g : α →ₘ[μ] ℝ≥0∞}
Goal: f ≤ g → f.lintegral ≤ g.lintegral
-- MeasureTheory.AEEqFun.coeFn_abs
theorem coeFn_abs {β} [TopologicalSpace β] [Lattice β] [TopologicalLattice β] [AddGroup β]
    [TopologicalAddGroup β] (f : α →ₘ[μ] β) : ⇑|f| =ᵐ[μ] fun x => |f x| 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {β : Type u_5} [TopologicalSpace β] [Lattice β] [TopologicalLattice β] [AddGroup β] [TopologicalAddGroup β] (f : α →ₘ[μ] β)
Goal: ↑|f| =ᵐ[μ] fun x => |↑f x|
-- MeasureTheory.AEEqFun.posPart_mk
@[simp]
theorem posPart_mk (f : α → γ) (hf) :
    posPart (mk f hf : α →ₘ[μ] γ) =
      mk (fun x => max (f x) 0)
        ((continuous_id.max continuous_const).comp_aestronglyMeasurable hf) 
Arguments: {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] [LinearOrder γ] [OrderClosedTopology γ] [Zero γ] (f : α → γ) (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: (MeasureTheory.AEEqFun.mk f hf).posPart = MeasureTheory.AEEqFun.mk (fun x => max (f x) 0) ⋯
-- MeasureTheory.AEEqFun.coeFn_posPart
theorem coeFn_posPart (f : α →ₘ[μ] γ) : ⇑(posPart f) =ᵐ[μ] fun a => max (f a) 0 
Arguments: {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace γ] [LinearOrder γ] [OrderClosedTopology γ] [Zero γ] (f : α →ₘ[μ] γ)
Goal: ↑f.posPart =ᵐ[μ] fun a => max (↑f a) 0
-- ContinuousMap.coeFn_toAEEqFun
theorem coeFn_toAEEqFun (f : C(α, β)) : f.toAEEqFun μ =ᵐ[μ] f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] (μ : MeasureTheory.Measure α) [TopologicalSpace α] [BorelSpace α] [TopologicalSpace β] [SecondCountableTopologyEither α β] [TopologicalSpace.PseudoMetrizableSpace β] (f : C(α, β))
Goal: ↑(ContinuousMap.toAEEqFun μ f) =ᵐ[μ] ⇑f
-- aeSeq.mk_eq_fun_of_mem_aeSeqSet
theorem mk_eq_fun_of_mem_aeSeqSet (hf : ∀ i, AEMeasurable (f i) μ) {x : α} (hx : x ∈ aeSeqSet hf p)
    (i : ι) : (hf i).mk (f i) x = f i x 
Arguments: {ι : Sort u_1} {α : Type u_2} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {f : ι → α → β} {μ : MeasureTheory.Measure α} {p : α → (ι → β) → Prop} (hf : ∀ (i : ι), AEMeasurable (f i) μ) {x : α} (hx : x ∈ aeSeqSet hf p) (i : ι)
Goal: AEMeasurable.mk (f i) ⋯ x = f i x
-- aeSeq.aeSeq_eq_mk_of_mem_aeSeqSet
theorem aeSeq_eq_mk_of_mem_aeSeqSet (hf : ∀ i, AEMeasurable (f i) μ) {x : α}
    (hx : x ∈ aeSeqSet hf p) (i : ι) : aeSeq hf p i x = (hf i).mk (f i) x 
Arguments: {ι : Sort u_1} {α : Type u_2} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {f : ι → α → β} {μ : MeasureTheory.Measure α} {p : α → (ι → β) → Prop} (hf : ∀ (i : ι), AEMeasurable (f i) μ) {x : α} (hx : x ∈ aeSeqSet hf p) (i : ι)
Goal: aeSeq hf p i x = AEMeasurable.mk (f i) ⋯ x
-- aeSeq.aeSeq_eq_fun_of_mem_aeSeqSet
theorem aeSeq_eq_fun_of_mem_aeSeqSet (hf : ∀ i, AEMeasurable (f i) μ) {x : α}
    (hx : x ∈ aeSeqSet hf p) (i : ι) : aeSeq hf p i x = f i x 
Arguments: {ι : Sort u_1} {α : Type u_2} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {f : ι → α → β} {μ : MeasureTheory.Measure α} {p : α → (ι → β) → Prop} (hf : ∀ (i : ι), AEMeasurable (f i) μ) {x : α} (hx : x ∈ aeSeqSet hf p) (i : ι)
Goal: aeSeq hf p i x = f i x
-- aeSeq.prop_of_mem_aeSeqSet
theorem prop_of_mem_aeSeqSet (hf : ∀ i, AEMeasurable (f i) μ) {x : α} (hx : x ∈ aeSeqSet hf p) :
    p x fun n => aeSeq hf p n x 
Arguments: {ι : Sort u_1} {α : Type u_2} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {f : ι → α → β} {μ : MeasureTheory.Measure α} {p : α → (ι → β) → Prop} (hf : ∀ (i : ι), AEMeasurable (f i) μ) {x : α} (hx : x ∈ aeSeqSet hf p)
Goal: p x fun n => aeSeq hf p n x
-- aeSeq.fun_prop_of_mem_aeSeqSet
theorem fun_prop_of_mem_aeSeqSet (hf : ∀ i, AEMeasurable (f i) μ) {x : α} (hx : x ∈ aeSeqSet hf p) :
    p x fun n => f n x 
Arguments: {ι : Sort u_1} {α : Type u_2} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {f : ι → α → β} {μ : MeasureTheory.Measure α} {p : α → (ι → β) → Prop} (hf : ∀ (i : ι), AEMeasurable (f i) μ) {x : α} (hx : x ∈ aeSeqSet hf p)
Goal: p x fun n => f n x
-- aeSeq.aeSeqSet_measurableSet
theorem aeSeqSet_measurableSet {hf : ∀ i, AEMeasurable (f i) μ} : MeasurableSet (aeSeqSet hf p) 
Arguments: {ι : Sort u_1} {α : Type u_2} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {f : ι → α → β} {μ : MeasureTheory.Measure α} {p : α → (ι → β) → Prop} {hf : ∀ (i : ι), AEMeasurable (f i) μ}
Goal: MeasurableSet (aeSeqSet hf p)
-- aeSeq.measurable
theorem measurable (hf : ∀ i, AEMeasurable (f i) μ) (p : α → (ι → β) → Prop) (i : ι) :
    Measurable (aeSeq hf p i) 
Arguments: {ι : Sort u_1} {α : Type u_2} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {f : ι → α → β} {μ : MeasureTheory.Measure α} (hf : ∀ (i : ι), AEMeasurable (f i) μ) (p : α → (ι → β) → Prop) (i : ι)
Goal: Measurable (aeSeq hf p i)
-- aeSeq.measure_compl_aeSeqSet_eq_zero
theorem measure_compl_aeSeqSet_eq_zero [Countable ι] (hf : ∀ i, AEMeasurable (f i) μ)
    (hp : ∀ᵐ x ∂μ, p x fun n => f n x) : μ (aeSeqSet hf p)ᶜ = 0 
Arguments: {ι : Sort u_1} {α : Type u_2} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {f : ι → α → β} {μ : MeasureTheory.Measure α} {p : α → (ι → β) → Prop} [Countable ι] (hf : ∀ (i : ι), AEMeasurable (f i) μ) (hp : ∀ᵐ (x : α) ∂μ, p x fun n => f n x)
Goal: μ (aeSeqSet hf p)ᶜ = 0
-- aeSeq.aeSeq_eq_mk_ae
theorem aeSeq_eq_mk_ae [Countable ι] (hf : ∀ i, AEMeasurable (f i) μ)
    (hp : ∀ᵐ x ∂μ, p x fun n => f n x) : ∀ᵐ a : α ∂μ, ∀ i : ι, aeSeq hf p i a = (hf i).mk (f i) a 
Arguments: {ι : Sort u_1} {α : Type u_2} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {f : ι → α → β} {μ : MeasureTheory.Measure α} {p : α → (ι → β) → Prop} [Countable ι] (hf : ∀ (i : ι), AEMeasurable (f i) μ) (hp : ∀ᵐ (x : α) ∂μ, p x fun n => f n x)
Goal: ∀ᵐ (a : α) ∂μ, ∀ (i : ι), aeSeq hf p i a = AEMeasurable.mk (f i) ⋯ a
-- aeSeq.aeSeq_eq_fun_ae
theorem aeSeq_eq_fun_ae [Countable ι] (hf : ∀ i, AEMeasurable (f i) μ)
    (hp : ∀ᵐ x ∂μ, p x fun n => f n x) : ∀ᵐ a : α ∂μ, ∀ i : ι, aeSeq hf p i a = f i a 
Arguments: {ι : Sort u_1} {α : Type u_2} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {f : ι → α → β} {μ : MeasureTheory.Measure α} {p : α → (ι → β) → Prop} [Countable ι] (hf : ∀ (i : ι), AEMeasurable (f i) μ) (hp : ∀ᵐ (x : α) ∂μ, p x fun n => f n x)
Goal: ∀ᵐ (a : α) ∂μ, ∀ (i : ι), aeSeq hf p i a = f i a
-- aeSeq.aeSeq_n_eq_fun_n_ae
theorem aeSeq_n_eq_fun_n_ae [Countable ι] (hf : ∀ i, AEMeasurable (f i) μ)
    (hp : ∀ᵐ x ∂μ, p x fun n => f n x) (n : ι) : aeSeq hf p n =ᵐ[μ] f n 
Arguments: {ι : Sort u_1} {α : Type u_2} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {f : ι → α → β} {μ : MeasureTheory.Measure α} {p : α → (ι → β) → Prop} [Countable ι] (hf : ∀ (i : ι), AEMeasurable (f i) μ) (hp : ∀ᵐ (x : α) ∂μ, p x fun n => f n x) (n : ι)
Goal: aeSeq hf p n =ᵐ[μ] f n
-- aeSeq.iSup
theorem iSup [CompleteLattice β] [Countable ι] (hf : ∀ i, AEMeasurable (f i) μ)
    (hp : ∀ᵐ x ∂μ, p x fun n => f n x) : ⨆ n, aeSeq hf p n =ᵐ[μ] ⨆ n, f n 
Arguments: {ι : Sort u_1} {α : Type u_2} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {f : ι → α → β} {μ : MeasureTheory.Measure α} {p : α → (ι → β) → Prop} [CompleteLattice β] [Countable ι] (hf : ∀ (i : ι), AEMeasurable (f i) μ) (hp : ∀ᵐ (x : α) ∂μ, p x fun n => f n x)
Goal: ⨆ n, aeSeq hf p n =ᵐ[μ] ⨆ n, f n
-- aeSeq.iInf
theorem iInf [CompleteLattice β] [Countable ι] (hf : ∀ i, AEMeasurable (f i) μ)
    (hp : ∀ᵐ x ∂μ, p x fun n ↦ f n x) : ⨅ n, aeSeq hf p n =ᵐ[μ] ⨅ n, f n 
Arguments: {ι : Sort u_1} {α : Type u_2} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {f : ι → α → β} {μ : MeasureTheory.Measure α} {p : α → (ι → β) → Prop} [CompleteLattice β] [Countable ι] (hf : ∀ (i : ι), AEMeasurable (f i) μ) (hp : ∀ᵐ (x : α) ∂μ, p x fun n => f n x)
Goal: ⨅ n, aeSeq hf p n =ᵐ[μ] ⨅ n, f n
-- MeasureTheory.tendsto_integral_of_dominated_convergence
theorem tendsto_integral_of_dominated_convergence {F : ℕ → α → G} {f : α → G} (bound : α → ℝ)
    (F_measurable : ∀ n, AEStronglyMeasurable (F n) μ) (bound_integrable : Integrable bound μ)
    (h_bound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a)
    (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))) :
    Tendsto (fun n => ∫ a, F n a ∂μ) atTop (𝓝 <| ∫ a, f a ∂μ) 
Arguments: {α : Type u_1} {G : Type u_3} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {F : ℕ → α → G} {f : α → G} (bound : α → ℝ) (F_measurable : ∀ (n : ℕ), MeasureTheory.AEStronglyMeasurable (F n) μ) (bound_integrable : MeasureTheory.Integrable bound μ) (h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a) (h_lim : ∀ᵐ (a : α) ∂μ, Filter.Tendsto (fun n => F n a) Filter.atTop (nhds (f a)))
Goal: Filter.Tendsto (fun n => ∫ (a : α), F n a ∂μ) Filter.atTop (nhds (∫ (a : α), f a ∂μ))
-- MeasureTheory.tendsto_integral_filter_of_dominated_convergence
theorem tendsto_integral_filter_of_dominated_convergence {ι} {l : Filter ι} [l.IsCountablyGenerated]
    {F : ι → α → G} {f : α → G} (bound : α → ℝ) (hF_meas : ∀ᶠ n in l, AEStronglyMeasurable (F n) μ)
    (h_bound : ∀ᶠ n in l, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a) (bound_integrable : Integrable bound μ)
    (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) l (𝓝 (f a))) :
    Tendsto (fun n => ∫ a, F n a ∂μ) l (𝓝 <| ∫ a, f a ∂μ) 
Arguments: {α : Type u_1} {G : Type u_3} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Type u_4} {l : Filter ι} [l.IsCountablyGenerated] {F : ι → α → G} {f : α → G} (bound : α → ℝ) (hF_meas : ∀ᶠ (n : ι) in l, MeasureTheory.AEStronglyMeasurable (F n) μ) (h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a) (bound_integrable : MeasureTheory.Integrable bound μ) (h_lim : ∀ᵐ (a : α) ∂μ, Filter.Tendsto (fun n => F n a) l (nhds (f a)))
Goal: Filter.Tendsto (fun n => ∫ (a : α), F n a ∂μ) l (nhds (∫ (a : α), f a ∂μ))
-- MeasureTheory.hasSum_integral_of_dominated_convergence
theorem hasSum_integral_of_dominated_convergence {ι} [Countable ι] {F : ι → α → G} {f : α → G}
    (bound : ι → α → ℝ) (hF_meas : ∀ n, AEStronglyMeasurable (F n) μ)
    (h_bound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound n a)
    (bound_summable : ∀ᵐ a ∂μ, Summable fun n => bound n a)
    (bound_integrable : Integrable (fun a => ∑' n, bound n a) μ)
    (h_lim : ∀ᵐ a ∂μ, HasSum (fun n => F n a) (f a)) :
    HasSum (fun n => ∫ a, F n a ∂μ) (∫ a, f a ∂μ) 
Arguments: {α : Type u_1} {G : Type u_3} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Type u_4} [Countable ι] {F : ι → α → G} {f : α → G} (bound : ι → α → ℝ) (hF_meas : ∀ (n : ι), MeasureTheory.AEStronglyMeasurable (F n) μ) (h_bound : ∀ (n : ι), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound n a) (bound_summable : ∀ᵐ (a : α) ∂μ, Summable fun n => bound n a) (bound_integrable : MeasureTheory.Integrable (fun a => ∑' (n : ι), bound n a) μ) (h_lim : ∀ᵐ (a : α) ∂μ, HasSum (fun n => F n a) (f a))
Goal: HasSum (fun n => ∫ (a : α), F n a ∂μ) (∫ (a : α), f a ∂μ)
-- MeasureTheory.integral_tsum
theorem integral_tsum {ι} [Countable ι] {f : ι → α → G} (hf : ∀ i, AEStronglyMeasurable (f i) μ)
    (hf' : ∑' i, ∫⁻ a : α, ‖f i a‖₊ ∂μ ≠ ∞) :
    ∫ a : α, ∑' i, f i a ∂μ = ∑' i, ∫ a : α, f i a ∂μ 
Arguments: {α : Type u_1} {G : Type u_3} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Type u_4} [Countable ι] {f : ι → α → G} (hf : ∀ (i : ι), MeasureTheory.AEStronglyMeasurable (f i) μ) (hf' : ∑' (i : ι), ∫⁻ (a : α), ↑‖f i a‖₊ ∂μ ≠ ⊤)
Goal: ∫ (a : α), ∑' (i : ι), f i a ∂μ = ∑' (i : ι), ∫ (a : α), f i a ∂μ
-- MeasureTheory.hasSum_integral_of_summable_integral_norm
lemma hasSum_integral_of_summable_integral_norm {ι} [Countable ι] {F : ι → α → E}
    (hF_int : ∀ i : ι, Integrable (F i) μ) (hF_sum : Summable fun i ↦ ∫ a, ‖F i a‖ ∂μ) :
    HasSum (∫ a, F · a ∂μ) (∫ a, (∑' i, F i a) ∂μ) 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Type u_4} [Countable ι] {F : ι → α → E} (hF_int : ∀ (i : ι), MeasureTheory.Integrable (F i) μ) (hF_sum : Summable fun i => ∫ (a : α), ‖F i a‖ ∂μ)
Goal: HasSum (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), ∑' (i : ι), F i a ∂μ)
-- MeasureTheory.integral_tsum_of_summable_integral_norm
lemma integral_tsum_of_summable_integral_norm {ι} [Countable ι] {F : ι → α → E}
    (hF_int : ∀ i : ι, Integrable (F i) μ) (hF_sum : Summable fun i ↦ ∫ a, ‖F i a‖ ∂μ) :
    ∑' i, (∫ a, F i a ∂μ) = ∫ a, (∑' i, F i a) ∂μ 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Type u_4} [Countable ι] {F : ι → α → E} (hF_int : ∀ (i : ι), MeasureTheory.Integrable (F i) μ) (hF_sum : Summable fun i => ∫ (a : α), ‖F i a‖ ∂μ)
Goal: ∑' (i : ι), ∫ (a : α), F i a ∂μ = ∫ (a : α), ∑' (i : ι), F i a ∂μ
-- Antitone.tendsto_setIntegral
theorem _root_.Antitone.tendsto_setIntegral (hsm : ∀ i, MeasurableSet (s i)) (h_anti : Antitone s)
    (hfi : IntegrableOn f (s 0) μ) :
    Tendsto (fun i => ∫ a in s i, f a ∂μ) atTop (𝓝 (∫ a in ⋂ n, s n, f a ∂μ)) 
Arguments: {α : Type u_1} {E : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedSpace ℝ E] {s : ℕ → Set α} {f : α → E} (hsm : ∀ (i : ℕ), MeasurableSet (s i)) (h_anti : Antitone s) (hfi : MeasureTheory.IntegrableOn f (s 0) μ)
Goal: Filter.Tendsto (fun i => ∫ (a : α) in s i, f a ∂μ) Filter.atTop (nhds (∫ (a : α) in ⋂ n, s n, f a ∂μ))
-- intervalIntegral.tendsto_integral_filter_of_dominated_convergence
nonrec theorem tendsto_integral_filter_of_dominated_convergence {ι} {l : Filter ι}
    [l.IsCountablyGenerated] {F : ι → ℝ → E} (bound : ℝ → ℝ)
    (hF_meas : ∀ᶠ n in l, AEStronglyMeasurable (F n) (μ.restrict (Ι a b)))
    (h_bound : ∀ᶠ n in l, ∀ᵐ x ∂μ, x ∈ Ι a b → ‖F n x‖ ≤ bound x)
    (bound_integrable : IntervalIntegrable bound μ a b)
    (h_lim : ∀ᵐ x ∂μ, x ∈ Ι a b → Tendsto (fun n => F n x) l (𝓝 (f x))) :
    Tendsto (fun n => ∫ x in a..b, F n x ∂μ) l (𝓝 <| ∫ x in a..b, f x ∂μ) 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} {f : ℝ → E} {μ : MeasureTheory.Measure ℝ} {ι : Type u_5} {l : Filter ι} [l.IsCountablyGenerated] {F : ι → ℝ → E} (bound : ℝ → ℝ) (hF_meas : ∀ᶠ (n : ι) in l, MeasureTheory.AEStronglyMeasurable (F n) (μ.restrict (Ι a b))) (h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (x : ℝ) ∂μ, x ∈ Ι a b → ‖F n x‖ ≤ bound x) (bound_integrable : IntervalIntegrable bound μ a b) (h_lim : ∀ᵐ (x : ℝ) ∂μ, x ∈ Ι a b → Filter.Tendsto (fun n => F n x) l (nhds (f x)))
Goal: Filter.Tendsto (fun n => ∫ (x : ℝ) in a..b, F n x ∂μ) l (nhds (∫ (x : ℝ) in a..b, f x ∂μ))
-- intervalIntegral.hasSum_integral_of_dominated_convergence
nonrec theorem hasSum_integral_of_dominated_convergence {ι} [Countable ι] {F : ι → ℝ → E}
    (bound : ι → ℝ → ℝ) (hF_meas : ∀ n, AEStronglyMeasurable (F n) (μ.restrict (Ι a b)))
    (h_bound : ∀ n, ∀ᵐ t ∂μ, t ∈ Ι a b → ‖F n t‖ ≤ bound n t)
    (bound_summable : ∀ᵐ t ∂μ, t ∈ Ι a b → Summable fun n => bound n t)
    (bound_integrable : IntervalIntegrable (fun t => ∑' n, bound n t) μ a b)
    (h_lim : ∀ᵐ t ∂μ, t ∈ Ι a b → HasSum (fun n => F n t) (f t)) :
    HasSum (fun n => ∫ t in a..b, F n t ∂μ) (∫ t in a..b, f t ∂μ) 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} {f : ℝ → E} {μ : MeasureTheory.Measure ℝ} {ι : Type u_5} [Countable ι] {F : ι → ℝ → E} (bound : ι → ℝ → ℝ) (hF_meas : ∀ (n : ι), MeasureTheory.AEStronglyMeasurable (F n) (μ.restrict (Ι a b))) (h_bound : ∀ (n : ι), ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι a b → ‖F n t‖ ≤ bound n t) (bound_summable : ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι a b → Summable fun n => bound n t) (bound_integrable : IntervalIntegrable (fun t => ∑' (n : ι), bound n t) μ a b) (h_lim : ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι a b → HasSum (fun n => F n t) (f t))
Goal: HasSum (fun n => ∫ (t : ℝ) in a..b, F n t ∂μ) (∫ (t : ℝ) in a..b, f t ∂μ)
-- intervalIntegral.hasSum_intervalIntegral_of_summable_norm
theorem hasSum_intervalIntegral_of_summable_norm [Countable ι] {f : ι → C(ℝ, E)}
    (hf_sum : Summable fun i : ι => ‖(f i).restrict (⟨uIcc a b, isCompact_uIcc⟩ : Compacts ℝ)‖) :
    HasSum (fun i : ι => ∫ x in a..b, f i x) (∫ x in a..b, ∑' i : ι, f i x) 
Arguments: {ι : Type u_1} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} [Countable ι] {f : ι → C(ℝ, E)} (hf_sum : Summable fun i => ‖ContinuousMap.restrict (↑{ carrier := Set.uIcc a b, isCompact' := ⋯ }) (f i)‖)
Goal: HasSum (fun i => ∫ (x : ℝ) in a..b, (f i) x) (∫ (x : ℝ) in a..b, ∑' (i : ι), (f i) x)
-- intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm
theorem tsum_intervalIntegral_eq_of_summable_norm [Countable ι] {f : ι → C(ℝ, E)}
    (hf_sum : Summable fun i : ι => ‖(f i).restrict (⟨uIcc a b, isCompact_uIcc⟩ : Compacts ℝ)‖) :
    ∑' i : ι, ∫ x in a..b, f i x = ∫ x in a..b, ∑' i : ι, f i x 
Arguments: {ι : Type u_1} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} [Countable ι] {f : ι → C(ℝ, E)} (hf_sum : Summable fun i => ‖ContinuousMap.restrict (↑{ carrier := Set.uIcc a b, isCompact' := ⋯ }) (f i)‖)
Goal: ∑' (i : ι), ∫ (x : ℝ) in a..b, (f i) x = ∫ (x : ℝ) in a..b, ∑' (i : ι), (f i) x
-- intervalIntegral.continuousWithinAt_of_dominated_interval
theorem continuousWithinAt_of_dominated_interval {F : X → ℝ → E} {x₀ : X} {bound : ℝ → ℝ} {a b : ℝ}
    {s : Set X} (hF_meas : ∀ᶠ x in 𝓝[s] x₀, AEStronglyMeasurable (F x) (μ.restrict <| Ι a b))
    (h_bound : ∀ᶠ x in 𝓝[s] x₀, ∀ᵐ t ∂μ, t ∈ Ι a b → ‖F x t‖ ≤ bound t)
    (bound_integrable : IntervalIntegrable bound μ a b)
    (h_cont : ∀ᵐ t ∂μ, t ∈ Ι a b → ContinuousWithinAt (fun x => F x t) s x₀) :
    ContinuousWithinAt (fun x => ∫ t in a..b, F x t ∂μ) s x₀ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure ℝ} {X : Type u_5} [TopologicalSpace X] [FirstCountableTopology X] {F : X → ℝ → E} {x₀ : X} {bound : ℝ → ℝ} {a b : ℝ} {s : Set X} (hF_meas : ∀ᶠ (x : X) in nhdsWithin x₀ s, MeasureTheory.AEStronglyMeasurable (F x) (μ.restrict (Ι a b))) (h_bound : ∀ᶠ (x : X) in nhdsWithin x₀ s, ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι a b → ‖F x t‖ ≤ bound t) (bound_integrable : IntervalIntegrable bound μ a b) (h_cont : ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι a b → ContinuousWithinAt (fun x => F x t) s x₀)
Goal: ContinuousWithinAt (fun x => ∫ (t : ℝ) in a..b, F x t ∂μ) s x₀
-- intervalIntegral.continuousAt_of_dominated_interval
theorem continuousAt_of_dominated_interval {F : X → ℝ → E} {x₀ : X} {bound : ℝ → ℝ} {a b : ℝ}
    (hF_meas : ∀ᶠ x in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict <| Ι a b))
    (h_bound : ∀ᶠ x in 𝓝 x₀, ∀ᵐ t ∂μ, t ∈ Ι a b → ‖F x t‖ ≤ bound t)
    (bound_integrable : IntervalIntegrable bound μ a b)
    (h_cont : ∀ᵐ t ∂μ, t ∈ Ι a b → ContinuousAt (fun x => F x t) x₀) :
    ContinuousAt (fun x => ∫ t in a..b, F x t ∂μ) x₀ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure ℝ} {X : Type u_5} [TopologicalSpace X] [FirstCountableTopology X] {F : X → ℝ → E} {x₀ : X} {bound : ℝ → ℝ} {a b : ℝ} (hF_meas : ∀ᶠ (x : X) in nhds x₀, MeasureTheory.AEStronglyMeasurable (F x) (μ.restrict (Ι a b))) (h_bound : ∀ᶠ (x : X) in nhds x₀, ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι a b → ‖F x t‖ ≤ bound t) (bound_integrable : IntervalIntegrable bound μ a b) (h_cont : ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι a b → ContinuousAt (fun x => F x t) x₀)
Goal: ContinuousAt (fun x => ∫ (t : ℝ) in a..b, F x t ∂μ) x₀
-- intervalIntegral.continuous_of_dominated_interval
theorem continuous_of_dominated_interval {F : X → ℝ → E} {bound : ℝ → ℝ} {a b : ℝ}
    (hF_meas : ∀ x, AEStronglyMeasurable (F x) <| μ.restrict <| Ι a b)
    (h_bound : ∀ x, ∀ᵐ t ∂μ, t ∈ Ι a b → ‖F x t‖ ≤ bound t)
    (bound_integrable : IntervalIntegrable bound μ a b)
    (h_cont : ∀ᵐ t ∂μ, t ∈ Ι a b → Continuous fun x => F x t) :
    Continuous fun x => ∫ t in a..b, F x t ∂μ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure ℝ} {X : Type u_5} [TopologicalSpace X] [FirstCountableTopology X] {F : X → ℝ → E} {bound : ℝ → ℝ} {a b : ℝ} (hF_meas : ∀ (x : X), MeasureTheory.AEStronglyMeasurable (F x) (μ.restrict (Ι a b))) (h_bound : ∀ (x : X), ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι a b → ‖F x t‖ ≤ bound t) (bound_integrable : IntervalIntegrable bound μ a b) (h_cont : ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι a b → Continuous fun x => F x t)
Goal: Continuous fun x => ∫ (t : ℝ) in a..b, F x t ∂μ
-- intervalIntegral.continuousWithinAt_primitive
theorem continuousWithinAt_primitive (hb₀ : μ {b₀} = 0)
    (h_int : IntervalIntegrable f μ (min a b₁) (max a b₂)) :
    ContinuousWithinAt (fun b => ∫ x in a..b, f x ∂μ) (Icc b₁ b₂) b₀ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b₀ b₁ b₂ : ℝ} {μ : MeasureTheory.Measure ℝ} {f : ℝ → E} (hb₀ : μ {b₀} = 0) (h_int : IntervalIntegrable f μ (min a b₁) (max a b₂))
Goal: ContinuousWithinAt (fun b => ∫ (x : ℝ) in a..b, f x ∂μ) (Set.Icc b₁ b₂) b₀
-- intervalIntegral.continuousAt_parametric_primitive_of_dominated
theorem continuousAt_parametric_primitive_of_dominated [FirstCountableTopology X]
    {F : X → ℝ → E} (bound : ℝ → ℝ) (a b : ℝ)
    {a₀ b₀ : ℝ} {x₀ : X} (hF_meas : ∀ x, AEStronglyMeasurable (F x) (μ.restrict <| Ι a b))
    (h_bound : ∀ᶠ x in 𝓝 x₀, ∀ᵐ t ∂μ.restrict <| Ι a b, ‖F x t‖ ≤ bound t)
    (bound_integrable : IntervalIntegrable bound μ a b)
    (h_cont : ∀ᵐ t ∂μ.restrict <| Ι a b, ContinuousAt (fun x ↦ F x t) x₀) (ha₀ : a₀ ∈ Ioo a b)
    (hb₀ : b₀ ∈ Ioo a b) (hμb₀ : μ {b₀} = 0) :
    ContinuousAt (fun p : X × ℝ ↦ ∫ t : ℝ in a₀..p.2, F p.1 t ∂μ) (x₀, b₀) 
Arguments: {E : Type u_1} {X : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [TopologicalSpace X] {μ : MeasureTheory.Measure ℝ} [FirstCountableTopology X] {F : X → ℝ → E} (bound : ℝ → ℝ) (a b : ℝ) {a₀ b₀ : ℝ} {x₀ : X} (hF_meas : ∀ (x : X), MeasureTheory.AEStronglyMeasurable (F x) (μ.restrict (Ι a b))) (h_bound : ∀ᶠ (x : X) in nhds x₀, ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x t‖ ≤ bound t) (bound_integrable : IntervalIntegrable bound μ a b) (h_cont : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ContinuousAt (fun x => F x t) x₀) (ha₀ : a₀ ∈ Set.Ioo a b) (hb₀ : b₀ ∈ Set.Ioo a b) (hμb₀ : μ {b₀} = 0)
Goal: ContinuousAt (fun p => ∫ (t : ℝ) in a₀..p.2, F p.1 t ∂μ) (x₀, b₀)
-- intervalIntegral.continuousOn_primitive
theorem continuousOn_primitive (h_int : IntegrableOn f (Icc a b) μ) :
    ContinuousOn (fun x => ∫ t in Ioc a x, f t ∂μ) (Icc a b) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} {μ : MeasureTheory.Measure ℝ} {f : ℝ → E} [MeasureTheory.NoAtoms μ] (h_int : MeasureTheory.IntegrableOn f (Set.Icc a b) μ)
Goal: ContinuousOn (fun x => ∫ (t : ℝ) in Set.Ioc a x, f t ∂μ) (Set.Icc a b)
-- intervalIntegral.continuousOn_primitive_Icc
theorem continuousOn_primitive_Icc (h_int : IntegrableOn f (Icc a b) μ) :
    ContinuousOn (fun x => ∫ t in Icc a x, f t ∂μ) (Icc a b) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} {μ : MeasureTheory.Measure ℝ} {f : ℝ → E} [MeasureTheory.NoAtoms μ] (h_int : MeasureTheory.IntegrableOn f (Set.Icc a b) μ)
Goal: ContinuousOn (fun x => ∫ (t : ℝ) in Set.Icc a x, f t ∂μ) (Set.Icc a b)
-- intervalIntegral.continuousOn_primitive_interval'
theorem continuousOn_primitive_interval' (h_int : IntervalIntegrable f μ b₁ b₂)
    (ha : a ∈ [[b₁, b₂]]) : ContinuousOn (fun b => ∫ x in a..b, f x ∂μ) [[b₁, b₂]] 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b₁ b₂ : ℝ} {μ : MeasureTheory.Measure ℝ} {f : ℝ → E} [MeasureTheory.NoAtoms μ] (h_int : IntervalIntegrable f μ b₁ b₂) (ha : a ∈ Set.uIcc b₁ b₂)
Goal: ContinuousOn (fun b => ∫ (x : ℝ) in a..b, f x ∂μ) (Set.uIcc b₁ b₂)
-- intervalIntegral.continuousOn_primitive_interval
theorem continuousOn_primitive_interval (h_int : IntegrableOn f (uIcc a b) μ) :
    ContinuousOn (fun x => ∫ t in a..x, f t ∂μ) (uIcc a b) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} {μ : MeasureTheory.Measure ℝ} {f : ℝ → E} [MeasureTheory.NoAtoms μ] (h_int : MeasureTheory.IntegrableOn f (Set.uIcc a b) μ)
Goal: ContinuousOn (fun x => ∫ (t : ℝ) in a..x, f t ∂μ) (Set.uIcc a b)