l3lab/ntp-mathlib huggingface dataset:
/-- Lebesgue dominated convergence theorem for series. -/
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) := by
  have hb_nonneg : ∀ᵐ a ∂μ, ∀ n, 0 ≤ bound n a :=
    eventually_countable_forall.2 fun n => (h_bound n).mono fun a => (norm_nonneg _).trans
  have hb_le_tsum : ∀ n, bound n ≤ᵐ[μ] fun a => ∑' n, bound n a := by
    intro n
    filter_upwards [hb_nonneg, bound_summable]
      with _ ha0 ha_sum using le_tsum ha_sum _ fun i _ => ha0 i
  have hF_integrable : ∀ n, Integrable (F n) μ := by
    refine' fun n => bound_integrable.mono' (hF_meas n) _
    exact EventuallyLE.trans (h_bound n) (hb_le_tsum n)
  simp only [HasSum, ← integral_finset_sum _ fun n _ => hF_integrable n]
  refine' tendsto_integral_filter_of_dominated_convergence
      (fun a => ∑' n, bound n a) _ _ bound_integrable h_lim
  · 

Proof state:
case refine'_1
α : Type u_1
E : Type u_2
F✝ : Type u_3
𝕜 : Type u_4
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : SMulCommClass ℝ 𝕜 E
inst✝⁵ : NormedAddCommGroup F✝
inst✝⁴ : NormedSpace ℝ F✝
inst✝³ : CompleteSpace F✝
G : Type u_5
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
ι : Type u_6
inst✝ : Countable ι
F : ι → α → G
f : α → G
bound : ι → α → ℝ
hF_meas : ∀ (n : ι), 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 : Integrable fun a => ∑' (n : ι), bound n a
h_lim : ∀ᵐ (a : α) ∂μ, HasSum (fun n => F n a) (f a)
hb_nonneg : ∀ᵐ (a : α) ∂μ, ∀ (n : ι), 0 ≤ bound n a
hb_le_tsum : ∀ (n : ι), bound n ≤ᵐ[μ] fun a => ∑' (n : ι), bound n a
hF_integrable : ∀ (n : ι), Integrable (F n)
⊢ ∀ᶠ (n : Finset ι) in atTop, AEStronglyMeasurable (fun a => ∑ i in n, F i a) μ

Next tactic:
exact eventually_of_forall fun s => s.aestronglyMeasurable_sum fun n _ => hF_meas n
/-- Lebesgue dominated convergence theorem for series. -/
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) := by
  have hb_nonneg : ∀ᵐ a ∂μ, ∀ n, 0 ≤ bound n a :=
    eventually_countable_forall.2 fun n => (h_bound n).mono fun a => (norm_nonneg _).trans
  have hb_le_tsum : ∀ n, bound n ≤ᵐ[μ] fun a => ∑' n, bound n a := by
    intro n
    filter_upwards [hb_nonneg, bound_summable]
      with _ ha0 ha_sum using le_tsum ha_sum _ fun i _ => ha0 i
  have hF_integrable : ∀ n, Integrable (F n) μ := by
    refine' fun n => bound_integrable.mono' (hF_meas n) _
    exact EventuallyLE.trans (h_bound n) (hb_le_tsum n)
  simp only [HasSum, ← integral_finset_sum _ fun n _ => hF_integrable n]
  refine' tendsto_integral_filter_of_dominated_convergence
      (fun a => ∑' n, bound n a) _ _ bound_integrable h_lim
  · exact eventually_of_forall fun s => s.aestronglyMeasurable_sum fun n _ => hF_meas n
  · 

Proof state:
case refine'_2
α : Type u_1
E : Type u_2
F✝ : Type u_3
𝕜 : Type u_4
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : SMulCommClass ℝ 𝕜 E
inst✝⁵ : NormedAddCommGroup F✝
inst✝⁴ : NormedSpace ℝ F✝
inst✝³ : CompleteSpace F✝
G : Type u_5
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
ι : Type u_6
inst✝ : Countable ι
F : ι → α → G
f : α → G
bound : ι → α → ℝ
hF_meas : ∀ (n : ι), 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 : Integrable fun a => ∑' (n : ι), bound n a
h_lim : ∀ᵐ (a : α) ∂μ, HasSum (fun n => F n a) (f a)
hb_nonneg : ∀ᵐ (a : α) ∂μ, ∀ (n : ι), 0 ≤ bound n a
hb_le_tsum : ∀ (n : ι), bound n ≤ᵐ[μ] fun a => ∑' (n : ι), bound n a
hF_integrable : ∀ (n : ι), Integrable (F n)
⊢ ∀ᶠ (n : Finset ι) in atTop, ∀ᵐ (a : α) ∂μ, ‖∑ i in n, F i a‖ ≤ (fun a => ∑' (n : ι), bound n a) a

Next tactic:
refine' eventually_of_forall fun s => _
/-- Lebesgue dominated convergence theorem for series. -/
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) := by
  have hb_nonneg : ∀ᵐ a ∂μ, ∀ n, 0 ≤ bound n a :=
    eventually_countable_forall.2 fun n => (h_bound n).mono fun a => (norm_nonneg _).trans
  have hb_le_tsum : ∀ n, bound n ≤ᵐ[μ] fun a => ∑' n, bound n a := by
    intro n
    filter_upwards [hb_nonneg, bound_summable]
      with _ ha0 ha_sum using le_tsum ha_sum _ fun i _ => ha0 i
  have hF_integrable : ∀ n, Integrable (F n) μ := by
    refine' fun n => bound_integrable.mono' (hF_meas n) _
    exact EventuallyLE.trans (h_bound n) (hb_le_tsum n)
  simp only [HasSum, ← integral_finset_sum _ fun n _ => hF_integrable n]
  refine' tendsto_integral_filter_of_dominated_convergence
      (fun a => ∑' n, bound n a) _ _ bound_integrable h_lim
  · exact eventually_of_forall fun s => s.aestronglyMeasurable_sum fun n _ => hF_meas n
  · refine' eventually_of_forall fun s => _
    

Proof state:
case refine'_2
α : Type u_1
E : Type u_2
F✝ : Type u_3
𝕜 : Type u_4
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : SMulCommClass ℝ 𝕜 E
inst✝⁵ : NormedAddCommGroup F✝
inst✝⁴ : NormedSpace ℝ F✝
inst✝³ : CompleteSpace F✝
G : Type u_5
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
ι : Type u_6
inst✝ : Countable ι
F : ι → α → G
f : α → G
bound : ι → α → ℝ
hF_meas : ∀ (n : ι), 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 : Integrable fun a => ∑' (n : ι), bound n a
h_lim : ∀ᵐ (a : α) ∂μ, HasSum (fun n => F n a) (f a)
hb_nonneg : ∀ᵐ (a : α) ∂μ, ∀ (n : ι), 0 ≤ bound n a
hb_le_tsum : ∀ (n : ι), bound n ≤ᵐ[μ] fun a => ∑' (n : ι), bound n a
hF_integrable : ∀ (n : ι), Integrable (F n)
s : Finset ι
⊢ ∀ᵐ (a : α) ∂μ, ‖∑ i in s, F i a‖ ≤ (fun a => ∑' (n : ι), bound n a) a

Next tactic:
filter_upwards [eventually_countable_forall.2 h_bound, hb_nonneg, bound_summable]
      with a hFa ha0 has
/-- Lebesgue dominated convergence theorem for series. -/
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) := by
  have hb_nonneg : ∀ᵐ a ∂μ, ∀ n, 0 ≤ bound n a :=
    eventually_countable_forall.2 fun n => (h_bound n).mono fun a => (norm_nonneg _).trans
  have hb_le_tsum : ∀ n, bound n ≤ᵐ[μ] fun a => ∑' n, bound n a := by
    intro n
    filter_upwards [hb_nonneg, bound_summable]
      with _ ha0 ha_sum using le_tsum ha_sum _ fun i _ => ha0 i
  have hF_integrable : ∀ n, Integrable (F n) μ := by
    refine' fun n => bound_integrable.mono' (hF_meas n) _
    exact EventuallyLE.trans (h_bound n) (hb_le_tsum n)
  simp only [HasSum, ← integral_finset_sum _ fun n _ => hF_integrable n]
  refine' tendsto_integral_filter_of_dominated_convergence
      (fun a => ∑' n, bound n a) _ _ bound_integrable h_lim
  · exact eventually_of_forall fun s => s.aestronglyMeasurable_sum fun n _ => hF_meas n
  · refine' eventually_of_forall fun s => _
    filter_upwards [eventually_countable_forall.2 h_bound, hb_nonneg, bound_summable]
      with a hFa ha0 has
    

Proof state:
case h
α : Type u_1
E : Type u_2
F✝ : Type u_3
𝕜 : Type u_4
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : SMulCommClass ℝ 𝕜 E
inst✝⁵ : NormedAddCommGroup F✝
inst✝⁴ : NormedSpace ℝ F✝
inst✝³ : CompleteSpace F✝
G : Type u_5
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
ι : Type u_6
inst✝ : Countable ι
F : ι → α → G
f : α → G
bound : ι → α → ℝ
hF_meas : ∀ (n : ι), 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 : Integrable fun a => ∑' (n : ι), bound n a
h_lim : ∀ᵐ (a : α) ∂μ, HasSum (fun n => F n a) (f a)
hb_nonneg : ∀ᵐ (a : α) ∂μ, ∀ (n : ι), 0 ≤ bound n a
hb_le_tsum : ∀ (n : ι), bound n ≤ᵐ[μ] fun a => ∑' (n : ι), bound n a
hF_integrable : ∀ (n : ι), Integrable (F n)
s : Finset ι
a : α
hFa : ∀ (i : ι), ‖F i a‖ ≤ bound i a
ha0 : ∀ (n : ι), 0 ≤ bound n a
has : Summable fun n => bound n a
⊢ ‖∑ i in s, F i a‖ ≤ ∑' (n : ι), bound n a

Next tactic:
calc
      ‖∑ n in s, F n a‖ ≤ ∑ n in s, bound n a := norm_sum_le_of_le _ fun n _ => hFa n
      _ ≤ ∑' n, bound n a := sum_le_tsum _ (fun n _ => ha0 n) has
theorem continuousWithinAt_of_dominated {F : X → α → G} {x₀ : X} {bound : α → ℝ} {s : Set X}
    (hF_meas : ∀ᶠ x in 𝓝[s] x₀, AEStronglyMeasurable (F x) μ)
    (h_bound : ∀ᶠ x in 𝓝[s] x₀, ∀ᵐ a ∂μ, ‖F x a‖ ≤ bound a) (bound_integrable : Integrable bound μ)
    (h_cont : ∀ᵐ a ∂μ, ContinuousWithinAt (fun x => F x a) s x₀) :
    ContinuousWithinAt (fun x => ∫ a, F x a ∂μ) s x₀ := by
  

Proof state:
α : Type u_1
E : Type u_2
F✝ : Type u_3
𝕜 : Type u_4
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : SMulCommClass ℝ 𝕜 E
inst✝⁶ : NormedAddCommGroup F✝
inst✝⁵ : NormedSpace ℝ F✝
inst✝⁴ : CompleteSpace F✝
G : Type u_5
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
f g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
F : X → α → G
x₀ : X
bound : α → ℝ
s : Set X
hF_meas : ∀ᶠ (x : X) in 𝓝[s] x₀, AEStronglyMeasurable (F x) μ
h_bound : ∀ᶠ (x : X) in 𝓝[s] x₀, ∀ᵐ (a : α) ∂μ, ‖F x a‖ ≤ bound a
bound_integrable : Integrable bound
h_cont : ∀ᵐ (a : α) ∂μ, ContinuousWithinAt (fun x => F x a) s x₀
⊢ ContinuousWithinAt (fun x => ∫ (a : α), F x a ∂μ) s x₀

Next tactic:
by_cases hG : CompleteSpace G
theorem continuousWithinAt_of_dominated {F : X → α → G} {x₀ : X} {bound : α → ℝ} {s : Set X}
    (hF_meas : ∀ᶠ x in 𝓝[s] x₀, AEStronglyMeasurable (F x) μ)
    (h_bound : ∀ᶠ x in 𝓝[s] x₀, ∀ᵐ a ∂μ, ‖F x a‖ ≤ bound a) (bound_integrable : Integrable bound μ)
    (h_cont : ∀ᵐ a ∂μ, ContinuousWithinAt (fun x => F x a) s x₀) :
    ContinuousWithinAt (fun x => ∫ a, F x a ∂μ) s x₀ := by
  by_cases hG : CompleteSpace G
  · 

Proof state:
case pos
α : Type u_1
E : Type u_2
F✝ : Type u_3
𝕜 : Type u_4
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : SMulCommClass ℝ 𝕜 E
inst✝⁶ : NormedAddCommGroup F✝
inst✝⁵ : NormedSpace ℝ F✝
inst✝⁴ : CompleteSpace F✝
G : Type u_5
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
f g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
F : X → α → G
x₀ : X
bound : α → ℝ
s : Set X
hF_meas : ∀ᶠ (x : X) in 𝓝[s] x₀, AEStronglyMeasurable (F x) μ
h_bound : ∀ᶠ (x : X) in 𝓝[s] x₀, ∀ᵐ (a : α) ∂μ, ‖F x a‖ ≤ bound a
bound_integrable : Integrable bound
h_cont : ∀ᵐ (a : α) ∂μ, ContinuousWithinAt (fun x => F x a) s x₀
hG : CompleteSpace G
⊢ ContinuousWithinAt (fun x => ∫ (a : α), F x a ∂μ) s x₀

Next tactic:
simp only [integral, hG, L1.integral]
theorem continuousWithinAt_of_dominated {F : X → α → G} {x₀ : X} {bound : α → ℝ} {s : Set X}
    (hF_meas : ∀ᶠ x in 𝓝[s] x₀, AEStronglyMeasurable (F x) μ)
    (h_bound : ∀ᶠ x in 𝓝[s] x₀, ∀ᵐ a ∂μ, ‖F x a‖ ≤ bound a) (bound_integrable : Integrable bound μ)
    (h_cont : ∀ᵐ a ∂μ, ContinuousWithinAt (fun x => F x a) s x₀) :
    ContinuousWithinAt (fun x => ∫ a, F x a ∂μ) s x₀ := by
  by_cases hG : CompleteSpace G
  · simp only [integral, hG, L1.integral]
    

Proof state:
case pos
α : Type u_1
E : Type u_2
F✝ : Type u_3
𝕜 : Type u_4
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : SMulCommClass ℝ 𝕜 E
inst✝⁶ : NormedAddCommGroup F✝
inst✝⁵ : NormedSpace ℝ F✝
inst✝⁴ : CompleteSpace F✝
G : Type u_5
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
f g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
F : X → α → G
x₀ : X
bound : α → ℝ
s : Set X
hF_meas : ∀ᶠ (x : X) in 𝓝[s] x₀, AEStronglyMeasurable (F x) μ
h_bound : ∀ᶠ (x : X) in 𝓝[s] x₀, ∀ᵐ (a : α) ∂μ, ‖F x a‖ ≤ bound a
bound_integrable : Integrable bound
h_cont : ∀ᵐ (a : α) ∂μ, ContinuousWithinAt (fun x => F x a) s x₀
hG : CompleteSpace G
⊢ ContinuousWithinAt
    (fun x =>
      if h : True then
        if hf : Integrable fun a => F x a then L1.integralCLM (Integrable.toL1 (fun a => F x a) hf) else 0
      else 0)
    s x₀

Next tactic:
exact continuousWithinAt_setToFun_of_dominated (dominatedFinMeasAdditive_weightedSMul μ)
      hF_meas h_bound bound_integrable h_cont
theorem continuousWithinAt_of_dominated {F : X → α → G} {x₀ : X} {bound : α → ℝ} {s : Set X}
    (hF_meas : ∀ᶠ x in 𝓝[s] x₀, AEStronglyMeasurable (F x) μ)
    (h_bound : ∀ᶠ x in 𝓝[s] x₀, ∀ᵐ a ∂μ, ‖F x a‖ ≤ bound a) (bound_integrable : Integrable bound μ)
    (h_cont : ∀ᵐ a ∂μ, ContinuousWithinAt (fun x => F x a) s x₀) :
    ContinuousWithinAt (fun x => ∫ a, F x a ∂μ) s x₀ := by
  by_cases hG : CompleteSpace G
  · simp only [integral, hG, L1.integral]
    exact continuousWithinAt_setToFun_of_dominated (dominatedFinMeasAdditive_weightedSMul μ)
      hF_meas h_bound bound_integrable h_cont
  · 

Proof state:
case neg
α : Type u_1
E : Type u_2
F✝ : Type u_3
𝕜 : Type u_4
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : SMulCommClass ℝ 𝕜 E
inst✝⁶ : NormedAddCommGroup F✝
inst✝⁵ : NormedSpace ℝ F✝
inst✝⁴ : CompleteSpace F✝
G : Type u_5
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
f g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
F : X → α → G
x₀ : X
bound : α → ℝ
s : Set X
hF_meas : ∀ᶠ (x : X) in 𝓝[s] x₀, AEStronglyMeasurable (F x) μ
h_bound : ∀ᶠ (x : X) in 𝓝[s] x₀, ∀ᵐ (a : α) ∂μ, ‖F x a‖ ≤ bound a
bound_integrable : Integrable bound
h_cont : ∀ᵐ (a : α) ∂μ, ContinuousWithinAt (fun x => F x a) s x₀
hG : ¬CompleteSpace G
⊢ ContinuousWithinAt (fun x => ∫ (a : α), F x a ∂μ) s x₀

Next tactic:
simp [integral, hG, continuousWithinAt_const]
theorem continuousAt_of_dominated {F : X → α → G} {x₀ : X} {bound : α → ℝ}
    (hF_meas : ∀ᶠ x in 𝓝 x₀, AEStronglyMeasurable (F x) μ)
    (h_bound : ∀ᶠ x in 𝓝 x₀, ∀ᵐ a ∂μ, ‖F x a‖ ≤ bound a) (bound_integrable : Integrable bound μ)
    (h_cont : ∀ᵐ a ∂μ, ContinuousAt (fun x => F x a) x₀) :
    ContinuousAt (fun x => ∫ a, F x a ∂μ) x₀ := by
  

Proof state:
α : Type u_1
E : Type u_2
F✝ : Type u_3
𝕜 : Type u_4
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : SMulCommClass ℝ 𝕜 E
inst✝⁶ : NormedAddCommGroup F✝
inst✝⁵ : NormedSpace ℝ F✝
inst✝⁴ : CompleteSpace F✝
G : Type u_5
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
f g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
F : X → α → G
x₀ : X
bound : α → ℝ
hF_meas : ∀ᶠ (x : X) in 𝓝 x₀, AEStronglyMeasurable (F x) μ
h_bound : ∀ᶠ (x : X) in 𝓝 x₀, ∀ᵐ (a : α) ∂μ, ‖F x a‖ ≤ bound a
bound_integrable : Integrable bound
h_cont : ∀ᵐ (a : α) ∂μ, ContinuousAt (fun x => F x a) x₀
⊢ ContinuousAt (fun x => ∫ (a : α), F x a ∂μ) x₀

Next tactic:
by_cases hG : CompleteSpace G
theorem continuousAt_of_dominated {F : X → α → G} {x₀ : X} {bound : α → ℝ}
    (hF_meas : ∀ᶠ x in 𝓝 x₀, AEStronglyMeasurable (F x) μ)
    (h_bound : ∀ᶠ x in 𝓝 x₀, ∀ᵐ a ∂μ, ‖F x a‖ ≤ bound a) (bound_integrable : Integrable bound μ)
    (h_cont : ∀ᵐ a ∂μ, ContinuousAt (fun x => F x a) x₀) :
    ContinuousAt (fun x => ∫ a, F x a ∂μ) x₀ := by
  by_cases hG : CompleteSpace G
  · 

Proof state:
case pos
α : Type u_1
E : Type u_2
F✝ : Type u_3
𝕜 : Type u_4
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : SMulCommClass ℝ 𝕜 E
inst✝⁶ : NormedAddCommGroup F✝
inst✝⁵ : NormedSpace ℝ F✝
inst✝⁴ : CompleteSpace F✝
G : Type u_5
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
f g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
F : X → α → G
x₀ : X
bound : α → ℝ
hF_meas : ∀ᶠ (x : X) in 𝓝 x₀, AEStronglyMeasurable (F x) μ
h_bound : ∀ᶠ (x : X) in 𝓝 x₀, ∀ᵐ (a : α) ∂μ, ‖F x a‖ ≤ bound a
bound_integrable : Integrable bound
h_cont : ∀ᵐ (a : α) ∂μ, ContinuousAt (fun x => F x a) x₀
hG : CompleteSpace G
⊢ ContinuousAt (fun x => ∫ (a : α), F x a ∂μ) x₀

Next tactic:
simp only [integral, hG, L1.integral]
theorem continuousAt_of_dominated {F : X → α → G} {x₀ : X} {bound : α → ℝ}
    (hF_meas : ∀ᶠ x in 𝓝 x₀, AEStronglyMeasurable (F x) μ)
    (h_bound : ∀ᶠ x in 𝓝 x₀, ∀ᵐ a ∂μ, ‖F x a‖ ≤ bound a) (bound_integrable : Integrable bound μ)
    (h_cont : ∀ᵐ a ∂μ, ContinuousAt (fun x => F x a) x₀) :
    ContinuousAt (fun x => ∫ a, F x a ∂μ) x₀ := by
  by_cases hG : CompleteSpace G
  · simp only [integral, hG, L1.integral]
    

Proof state:
case pos
α : Type u_1
E : Type u_2
F✝ : Type u_3
𝕜 : Type u_4
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : SMulCommClass ℝ 𝕜 E
inst✝⁶ : NormedAddCommGroup F✝
inst✝⁵ : NormedSpace ℝ F✝
inst✝⁴ : CompleteSpace F✝
G : Type u_5
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
f g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
F : X → α → G
x₀ : X
bound : α → ℝ
hF_meas : ∀ᶠ (x : X) in 𝓝 x₀, AEStronglyMeasurable (F x) μ
h_bound : ∀ᶠ (x : X) in 𝓝 x₀, ∀ᵐ (a : α) ∂μ, ‖F x a‖ ≤ bound a
bound_integrable : Integrable bound
h_cont : ∀ᵐ (a : α) ∂μ, ContinuousAt (fun x => F x a) x₀
hG : CompleteSpace G
⊢ ContinuousAt
    (fun x =>
      if h : True then
        if hf : Integrable fun a => F x a then L1.integralCLM (Integrable.toL1 (fun a => F x a) hf) else 0
      else 0)
    x₀

Next tactic:
exact continuousAt_setToFun_of_dominated (dominatedFinMeasAdditive_weightedSMul μ)
      hF_meas h_bound bound_integrable h_cont
theorem continuousAt_of_dominated {F : X → α → G} {x₀ : X} {bound : α → ℝ}
    (hF_meas : ∀ᶠ x in 𝓝 x₀, AEStronglyMeasurable (F x) μ)
    (h_bound : ∀ᶠ x in 𝓝 x₀, ∀ᵐ a ∂μ, ‖F x a‖ ≤ bound a) (bound_integrable : Integrable bound μ)
    (h_cont : ∀ᵐ a ∂μ, ContinuousAt (fun x => F x a) x₀) :
    ContinuousAt (fun x => ∫ a, F x a ∂μ) x₀ := by
  by_cases hG : CompleteSpace G
  · simp only [integral, hG, L1.integral]
    exact continuousAt_setToFun_of_dominated (dominatedFinMeasAdditive_weightedSMul μ)
      hF_meas h_bound bound_integrable h_cont
  · 

Proof state:
case neg
α : Type u_1
E : Type u_2
F✝ : Type u_3
𝕜 : Type u_4
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : SMulCommClass ℝ 𝕜 E
inst✝⁶ : NormedAddCommGroup F✝
inst✝⁵ : NormedSpace ℝ F✝
inst✝⁴ : CompleteSpace F✝
G : Type u_5
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
f g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
F : X → α → G
x₀ : X
bound : α → ℝ
hF_meas : ∀ᶠ (x : X) in 𝓝 x₀, AEStronglyMeasurable (F x) μ
h_bound : ∀ᶠ (x : X) in 𝓝 x₀, ∀ᵐ (a : α) ∂μ, ‖F x a‖ ≤ bound a
bound_integrable : Integrable bound
h_cont : ∀ᵐ (a : α) ∂μ, ContinuousAt (fun x => F x a) x₀
hG : ¬CompleteSpace G
⊢ ContinuousAt (fun x => ∫ (a : α), F x a ∂μ) x₀

Next tactic:
simp [integral, hG, continuousAt_const]
theorem continuousOn_of_dominated {F : X → α → G} {bound : α → ℝ} {s : Set X}
    (hF_meas : ∀ x ∈ s, AEStronglyMeasurable (F x) μ)
    (h_bound : ∀ x ∈ s, ∀ᵐ a ∂μ, ‖F x a‖ ≤ bound a) (bound_integrable : Integrable bound μ)
    (h_cont : ∀ᵐ a ∂μ, ContinuousOn (fun x => F x a) s) :
    ContinuousOn (fun x => ∫ a, F x a ∂μ) s := by
  

Proof state:
α : Type u_1
E : Type u_2
F✝ : Type u_3
𝕜 : Type u_4
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : SMulCommClass ℝ 𝕜 E
inst✝⁶ : NormedAddCommGroup F✝
inst✝⁵ : NormedSpace ℝ F✝
inst✝⁴ : CompleteSpace F✝
G : Type u_5
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
f g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
F : X → α → G
bound : α → ℝ
s : Set X
hF_meas : ∀ x ∈ s, AEStronglyMeasurable (F x) μ
h_bound : ∀ x ∈ s, ∀ᵐ (a : α) ∂μ, ‖F x a‖ ≤ bound a
bound_integrable : Integrable bound
h_cont : ∀ᵐ (a : α) ∂μ, ContinuousOn (fun x => F x a) s
⊢ ContinuousOn (fun x => ∫ (a : α), F x a ∂μ) s

Next tactic:
by_cases hG : CompleteSpace G
theorem continuousOn_of_dominated {F : X → α → G} {bound : α → ℝ} {s : Set X}
    (hF_meas : ∀ x ∈ s, AEStronglyMeasurable (F x) μ)
    (h_bound : ∀ x ∈ s, ∀ᵐ a ∂μ, ‖F x a‖ ≤ bound a) (bound_integrable : Integrable bound μ)
    (h_cont : ∀ᵐ a ∂μ, ContinuousOn (fun x => F x a) s) :
    ContinuousOn (fun x => ∫ a, F x a ∂μ) s := by
  by_cases hG : CompleteSpace G
  · 

Proof state:
case pos
α : Type u_1
E : Type u_2
F✝ : Type u_3
𝕜 : Type u_4
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : SMulCommClass ℝ 𝕜 E
inst✝⁶ : NormedAddCommGroup F✝
inst✝⁵ : NormedSpace ℝ F✝
inst✝⁴ : CompleteSpace F✝
G : Type u_5
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
f g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
F : X → α → G
bound : α → ℝ
s : Set X
hF_meas : ∀ x ∈ s, AEStronglyMeasurable (F x) μ
h_bound : ∀ x ∈ s, ∀ᵐ (a : α) ∂μ, ‖F x a‖ ≤ bound a
bound_integrable : Integrable bound
h_cont : ∀ᵐ (a : α) ∂μ, ContinuousOn (fun x => F x a) s
hG : CompleteSpace G
⊢ ContinuousOn (fun x => ∫ (a : α), F x a ∂μ) s

Next tactic:
simp only [integral, hG, L1.integral]