l3lab/ntp-mathlib huggingface dataset:
/-- If `F i → f` in `L1`, then `setToFun μ T hT (F i) → setToFun μ T hT f`. -/
theorem tendsto_setToFun_of_L1 (hT : DominatedFinMeasAdditive μ T C) {ι} (f : α → E)
    (hfi : Integrable f μ) {fs : ι → α → E} {l : Filter ι} (hfsi : ∀ᶠ i in l, Integrable (fs i) μ)
    (hfs : Tendsto (fun i => ∫⁻ x, ‖fs i x - f x‖₊ ∂μ) l (𝓝 0)) :
    Tendsto (fun i => setToFun μ T hT (fs i)) l (𝓝 <| setToFun μ T hT f) := by
  classical
    let f_lp := hfi.toL1 f
    let F_lp i := if hFi : Integrable (fs i) μ then hFi.toL1 (fs i) else 0
    have tendsto_L1 : Tendsto F_lp l (𝓝 f_lp) := by
      rw [Lp.tendsto_Lp_iff_tendsto_ℒp']
      

Proof state:
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
f : α → E
hfi : Integrable f
fs : ι → α → E
l : Filter ι
hfsi : ∀ᶠ (i : ι) in l, Integrable (fs i)
hfs : Tendsto (fun i => ∫⁻ (x : α), ↑‖fs i x - f x‖₊ ∂μ) l (𝓝 0)
f_lp : ↥(Lp E 1) := Integrable.toL1 f hfi
F_lp : ι → ↥(Lp E 1) := fun i => if hFi : Integrable (fs i) then Integrable.toL1 (fs i) hFi else 0
⊢ Tendsto (fun n => snorm (↑↑(F_lp n) - ↑↑f_lp) 1 μ) l (𝓝 0)

Next tactic:
simp_rw [snorm_one_eq_lintegral_nnnorm, Pi.sub_apply]
/-- If `F i → f` in `L1`, then `setToFun μ T hT (F i) → setToFun μ T hT f`. -/
theorem tendsto_setToFun_of_L1 (hT : DominatedFinMeasAdditive μ T C) {ι} (f : α → E)
    (hfi : Integrable f μ) {fs : ι → α → E} {l : Filter ι} (hfsi : ∀ᶠ i in l, Integrable (fs i) μ)
    (hfs : Tendsto (fun i => ∫⁻ x, ‖fs i x - f x‖₊ ∂μ) l (𝓝 0)) :
    Tendsto (fun i => setToFun μ T hT (fs i)) l (𝓝 <| setToFun μ T hT f) := by
  classical
    let f_lp := hfi.toL1 f
    let F_lp i := if hFi : Integrable (fs i) μ then hFi.toL1 (fs i) else 0
    have tendsto_L1 : Tendsto F_lp l (𝓝 f_lp) := by
      rw [Lp.tendsto_Lp_iff_tendsto_ℒp']
      simp_rw [snorm_one_eq_lintegral_nnnorm, Pi.sub_apply]
      

Proof state:
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
f : α → E
hfi : Integrable f
fs : ι → α → E
l : Filter ι
hfsi : ∀ᶠ (i : ι) in l, Integrable (fs i)
hfs : Tendsto (fun i => ∫⁻ (x : α), ↑‖fs i x - f x‖₊ ∂μ) l (𝓝 0)
f_lp : ↥(Lp E 1) := Integrable.toL1 f hfi
F_lp : ι → ↥(Lp E 1) := fun i => if hFi : Integrable (fs i) then Integrable.toL1 (fs i) hFi else 0
⊢ Tendsto
    (fun n =>
      ∫⁻ (x : α),
        ↑‖↑↑(if hFi : Integrable (fs n) then Integrable.toL1 (fs n) hFi else 0) x - ↑↑(Integrable.toL1 f hfi) x‖₊ ∂μ)
    l (𝓝 0)

Next tactic:
refine' (tendsto_congr' _).mp hfs
/-- If `F i → f` in `L1`, then `setToFun μ T hT (F i) → setToFun μ T hT f`. -/
theorem tendsto_setToFun_of_L1 (hT : DominatedFinMeasAdditive μ T C) {ι} (f : α → E)
    (hfi : Integrable f μ) {fs : ι → α → E} {l : Filter ι} (hfsi : ∀ᶠ i in l, Integrable (fs i) μ)
    (hfs : Tendsto (fun i => ∫⁻ x, ‖fs i x - f x‖₊ ∂μ) l (𝓝 0)) :
    Tendsto (fun i => setToFun μ T hT (fs i)) l (𝓝 <| setToFun μ T hT f) := by
  classical
    let f_lp := hfi.toL1 f
    let F_lp i := if hFi : Integrable (fs i) μ then hFi.toL1 (fs i) else 0
    have tendsto_L1 : Tendsto F_lp l (𝓝 f_lp) := by
      rw [Lp.tendsto_Lp_iff_tendsto_ℒp']
      simp_rw [snorm_one_eq_lintegral_nnnorm, Pi.sub_apply]
      refine' (tendsto_congr' _).mp hfs
      

Proof state:
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
f : α → E
hfi : Integrable f
fs : ι → α → E
l : Filter ι
hfsi : ∀ᶠ (i : ι) in l, Integrable (fs i)
hfs : Tendsto (fun i => ∫⁻ (x : α), ↑‖fs i x - f x‖₊ ∂μ) l (𝓝 0)
f_lp : ↥(Lp E 1) := Integrable.toL1 f hfi
F_lp : ι → ↥(Lp E 1) := fun i => if hFi : Integrable (fs i) then Integrable.toL1 (fs i) hFi else 0
⊢ (fun i => ∫⁻ (x : α), ↑‖fs i x - f x‖₊ ∂μ) =ᶠ[l] fun n =>
    ∫⁻ (x : α),
      ↑‖↑↑(if hFi : Integrable (fs n) then Integrable.toL1 (fs n) hFi else 0) x - ↑↑(Integrable.toL1 f hfi) x‖₊ ∂μ

Next tactic:
filter_upwards [hfsi] with i hi
/-- If `F i → f` in `L1`, then `setToFun μ T hT (F i) → setToFun μ T hT f`. -/
theorem tendsto_setToFun_of_L1 (hT : DominatedFinMeasAdditive μ T C) {ι} (f : α → E)
    (hfi : Integrable f μ) {fs : ι → α → E} {l : Filter ι} (hfsi : ∀ᶠ i in l, Integrable (fs i) μ)
    (hfs : Tendsto (fun i => ∫⁻ x, ‖fs i x - f x‖₊ ∂μ) l (𝓝 0)) :
    Tendsto (fun i => setToFun μ T hT (fs i)) l (𝓝 <| setToFun μ T hT f) := by
  classical
    let f_lp := hfi.toL1 f
    let F_lp i := if hFi : Integrable (fs i) μ then hFi.toL1 (fs i) else 0
    have tendsto_L1 : Tendsto F_lp l (𝓝 f_lp) := by
      rw [Lp.tendsto_Lp_iff_tendsto_ℒp']
      simp_rw [snorm_one_eq_lintegral_nnnorm, Pi.sub_apply]
      refine' (tendsto_congr' _).mp hfs
      filter_upwards [hfsi] with i hi
      

Proof state:
case h
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
f : α → E
hfi : Integrable f
fs : ι → α → E
l : Filter ι
hfsi : ∀ᶠ (i : ι) in l, Integrable (fs i)
hfs : Tendsto (fun i => ∫⁻ (x : α), ↑‖fs i x - f x‖₊ ∂μ) l (𝓝 0)
f_lp : ↥(Lp E 1) := Integrable.toL1 f hfi
F_lp : ι → ↥(Lp E 1) := fun i => if hFi : Integrable (fs i) then Integrable.toL1 (fs i) hFi else 0
i : ι
hi : Integrable (fs i)
⊢ ∫⁻ (x : α), ↑‖fs i x - f x‖₊ ∂μ =
    ∫⁻ (x : α),
      ↑‖↑↑(if hFi : Integrable (fs i) then Integrable.toL1 (fs i) hFi else 0) x - ↑↑(Integrable.toL1 f hfi) x‖₊ ∂μ

Next tactic:
refine' lintegral_congr_ae _
/-- If `F i → f` in `L1`, then `setToFun μ T hT (F i) → setToFun μ T hT f`. -/
theorem tendsto_setToFun_of_L1 (hT : DominatedFinMeasAdditive μ T C) {ι} (f : α → E)
    (hfi : Integrable f μ) {fs : ι → α → E} {l : Filter ι} (hfsi : ∀ᶠ i in l, Integrable (fs i) μ)
    (hfs : Tendsto (fun i => ∫⁻ x, ‖fs i x - f x‖₊ ∂μ) l (𝓝 0)) :
    Tendsto (fun i => setToFun μ T hT (fs i)) l (𝓝 <| setToFun μ T hT f) := by
  classical
    let f_lp := hfi.toL1 f
    let F_lp i := if hFi : Integrable (fs i) μ then hFi.toL1 (fs i) else 0
    have tendsto_L1 : Tendsto F_lp l (𝓝 f_lp) := by
      rw [Lp.tendsto_Lp_iff_tendsto_ℒp']
      simp_rw [snorm_one_eq_lintegral_nnnorm, Pi.sub_apply]
      refine' (tendsto_congr' _).mp hfs
      filter_upwards [hfsi] with i hi
      refine' lintegral_congr_ae _
      

Proof state:
case h
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
f : α → E
hfi : Integrable f
fs : ι → α → E
l : Filter ι
hfsi : ∀ᶠ (i : ι) in l, Integrable (fs i)
hfs : Tendsto (fun i => ∫⁻ (x : α), ↑‖fs i x - f x‖₊ ∂μ) l (𝓝 0)
f_lp : ↥(Lp E 1) := Integrable.toL1 f hfi
F_lp : ι → ↥(Lp E 1) := fun i => if hFi : Integrable (fs i) then Integrable.toL1 (fs i) hFi else 0
i : ι
hi : Integrable (fs i)
⊢ (fun x => ↑‖fs i x - f x‖₊) =ᵐ[μ] fun x =>
    ↑‖↑↑(if hFi : Integrable (fs i) then Integrable.toL1 (fs i) hFi else 0) x - ↑↑(Integrable.toL1 f hfi) x‖₊

Next tactic:
filter_upwards [hi.coeFn_toL1, hfi.coeFn_toL1] with x hxi hxf
/-- If `F i → f` in `L1`, then `setToFun μ T hT (F i) → setToFun μ T hT f`. -/
theorem tendsto_setToFun_of_L1 (hT : DominatedFinMeasAdditive μ T C) {ι} (f : α → E)
    (hfi : Integrable f μ) {fs : ι → α → E} {l : Filter ι} (hfsi : ∀ᶠ i in l, Integrable (fs i) μ)
    (hfs : Tendsto (fun i => ∫⁻ x, ‖fs i x - f x‖₊ ∂μ) l (𝓝 0)) :
    Tendsto (fun i => setToFun μ T hT (fs i)) l (𝓝 <| setToFun μ T hT f) := by
  classical
    let f_lp := hfi.toL1 f
    let F_lp i := if hFi : Integrable (fs i) μ then hFi.toL1 (fs i) else 0
    have tendsto_L1 : Tendsto F_lp l (𝓝 f_lp) := by
      rw [Lp.tendsto_Lp_iff_tendsto_ℒp']
      simp_rw [snorm_one_eq_lintegral_nnnorm, Pi.sub_apply]
      refine' (tendsto_congr' _).mp hfs
      filter_upwards [hfsi] with i hi
      refine' lintegral_congr_ae _
      filter_upwards [hi.coeFn_toL1, hfi.coeFn_toL1] with x hxi hxf
      

Proof state:
case h
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
f : α → E
hfi : Integrable f
fs : ι → α → E
l : Filter ι
hfsi : ∀ᶠ (i : ι) in l, Integrable (fs i)
hfs : Tendsto (fun i => ∫⁻ (x : α), ↑‖fs i x - f x‖₊ ∂μ) l (𝓝 0)
f_lp : ↥(Lp E 1) := Integrable.toL1 f hfi
F_lp : ι → ↥(Lp E 1) := fun i => if hFi : Integrable (fs i) then Integrable.toL1 (fs i) hFi else 0
i : ι
hi : Integrable (fs i)
x : α
hxi : ↑↑(Integrable.toL1 (fs i) hi) x = fs i x
hxf : ↑↑(Integrable.toL1 f hfi) x = f x
⊢ ↑‖fs i x - f x‖₊ =
    ↑‖↑↑(if hFi : Integrable (fs i) then Integrable.toL1 (fs i) hFi else 0) x - ↑↑(Integrable.toL1 f hfi) x‖₊

Next tactic:
simp_rw [dif_pos hi, hxi, hxf]
/-- If `F i → f` in `L1`, then `setToFun μ T hT (F i) → setToFun μ T hT f`. -/
theorem tendsto_setToFun_of_L1 (hT : DominatedFinMeasAdditive μ T C) {ι} (f : α → E)
    (hfi : Integrable f μ) {fs : ι → α → E} {l : Filter ι} (hfsi : ∀ᶠ i in l, Integrable (fs i) μ)
    (hfs : Tendsto (fun i => ∫⁻ x, ‖fs i x - f x‖₊ ∂μ) l (𝓝 0)) :
    Tendsto (fun i => setToFun μ T hT (fs i)) l (𝓝 <| setToFun μ T hT f) := by
  classical
    let f_lp := hfi.toL1 f
    let F_lp i := if hFi : Integrable (fs i) μ then hFi.toL1 (fs i) else 0
    have tendsto_L1 : Tendsto F_lp l (𝓝 f_lp) := by
      rw [Lp.tendsto_Lp_iff_tendsto_ℒp']
      simp_rw [snorm_one_eq_lintegral_nnnorm, Pi.sub_apply]
      refine' (tendsto_congr' _).mp hfs
      filter_upwards [hfsi] with i hi
      refine' lintegral_congr_ae _
      filter_upwards [hi.coeFn_toL1, hfi.coeFn_toL1] with x hxi hxf
      simp_rw [dif_pos hi, hxi, hxf]
    

Proof state:
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
f : α → E
hfi : Integrable f
fs : ι → α → E
l : Filter ι
hfsi : ∀ᶠ (i : ι) in l, Integrable (fs i)
hfs : Tendsto (fun i => ∫⁻ (x : α), ↑‖fs i x - f x‖₊ ∂μ) l (𝓝 0)
f_lp : ↥(Lp E 1) := Integrable.toL1 f hfi
F_lp : ι → ↥(Lp E 1) := fun i => if hFi : Integrable (fs i) then Integrable.toL1 (fs i) hFi else 0
tendsto_L1 : Tendsto F_lp l (𝓝 f_lp)
⊢ Tendsto (fun i => setToFun μ T hT (fs i)) l (𝓝 (setToFun μ T hT f))

Next tactic:
suffices Tendsto (fun i => setToFun μ T hT (F_lp i)) l (𝓝 (setToFun μ T hT f)) by
      refine' (tendsto_congr' _).mp this
      filter_upwards [hfsi] with i hi
      suffices h_ae_eq : F_lp i =ᵐ[μ] fs i; exact setToFun_congr_ae hT h_ae_eq
      simp_rw [dif_pos hi]
      exact hi.coeFn_toL1
/-- If `F i → f` in `L1`, then `setToFun μ T hT (F i) → setToFun μ T hT f`. -/
theorem tendsto_setToFun_of_L1 (hT : DominatedFinMeasAdditive μ T C) {ι} (f : α → E)
    (hfi : Integrable f μ) {fs : ι → α → E} {l : Filter ι} (hfsi : ∀ᶠ i in l, Integrable (fs i) μ)
    (hfs : Tendsto (fun i => ∫⁻ x, ‖fs i x - f x‖₊ ∂μ) l (𝓝 0)) :
    Tendsto (fun i => setToFun μ T hT (fs i)) l (𝓝 <| setToFun μ T hT f) := by
  classical
    let f_lp := hfi.toL1 f
    let F_lp i := if hFi : Integrable (fs i) μ then hFi.toL1 (fs i) else 0
    have tendsto_L1 : Tendsto F_lp l (𝓝 f_lp) := by
      rw [Lp.tendsto_Lp_iff_tendsto_ℒp']
      simp_rw [snorm_one_eq_lintegral_nnnorm, Pi.sub_apply]
      refine' (tendsto_congr' _).mp hfs
      filter_upwards [hfsi] with i hi
      refine' lintegral_congr_ae _
      filter_upwards [hi.coeFn_toL1, hfi.coeFn_toL1] with x hxi hxf
      simp_rw [dif_pos hi, hxi, hxf]
    suffices Tendsto (fun i => setToFun μ T hT (F_lp i)) l (𝓝 (setToFun μ T hT f)) by
      

Proof state:
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
f : α → E
hfi : Integrable f
fs : ι → α → E
l : Filter ι
hfsi : ∀ᶠ (i : ι) in l, Integrable (fs i)
hfs : Tendsto (fun i => ∫⁻ (x : α), ↑‖fs i x - f x‖₊ ∂μ) l (𝓝 0)
f_lp : ↥(Lp E 1) := Integrable.toL1 f hfi
F_lp : ι → ↥(Lp E 1) := fun i => if hFi : Integrable (fs i) then Integrable.toL1 (fs i) hFi else 0
tendsto_L1 : Tendsto F_lp l (𝓝 f_lp)
this : Tendsto (fun i => setToFun μ T hT ↑↑(F_lp i)) l (𝓝 (setToFun μ T hT f))
⊢ Tendsto (fun i => setToFun μ T hT (fs i)) l (𝓝 (setToFun μ T hT f))

Next tactic:
refine' (tendsto_congr' _).mp this
/-- If `F i → f` in `L1`, then `setToFun μ T hT (F i) → setToFun μ T hT f`. -/
theorem tendsto_setToFun_of_L1 (hT : DominatedFinMeasAdditive μ T C) {ι} (f : α → E)
    (hfi : Integrable f μ) {fs : ι → α → E} {l : Filter ι} (hfsi : ∀ᶠ i in l, Integrable (fs i) μ)
    (hfs : Tendsto (fun i => ∫⁻ x, ‖fs i x - f x‖₊ ∂μ) l (𝓝 0)) :
    Tendsto (fun i => setToFun μ T hT (fs i)) l (𝓝 <| setToFun μ T hT f) := by
  classical
    let f_lp := hfi.toL1 f
    let F_lp i := if hFi : Integrable (fs i) μ then hFi.toL1 (fs i) else 0
    have tendsto_L1 : Tendsto F_lp l (𝓝 f_lp) := by
      rw [Lp.tendsto_Lp_iff_tendsto_ℒp']
      simp_rw [snorm_one_eq_lintegral_nnnorm, Pi.sub_apply]
      refine' (tendsto_congr' _).mp hfs
      filter_upwards [hfsi] with i hi
      refine' lintegral_congr_ae _
      filter_upwards [hi.coeFn_toL1, hfi.coeFn_toL1] with x hxi hxf
      simp_rw [dif_pos hi, hxi, hxf]
    suffices Tendsto (fun i => setToFun μ T hT (F_lp i)) l (𝓝 (setToFun μ T hT f)) by
      refine' (tendsto_congr' _).mp this
      

Proof state:
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
f : α → E
hfi : Integrable f
fs : ι → α → E
l : Filter ι
hfsi : ∀ᶠ (i : ι) in l, Integrable (fs i)
hfs : Tendsto (fun i => ∫⁻ (x : α), ↑‖fs i x - f x‖₊ ∂μ) l (𝓝 0)
f_lp : ↥(Lp E 1) := Integrable.toL1 f hfi
F_lp : ι → ↥(Lp E 1) := fun i => if hFi : Integrable (fs i) then Integrable.toL1 (fs i) hFi else 0
tendsto_L1 : Tendsto F_lp l (𝓝 f_lp)
this : Tendsto (fun i => setToFun μ T hT ↑↑(F_lp i)) l (𝓝 (setToFun μ T hT f))
⊢ (fun i => setToFun μ T hT ↑↑(F_lp i)) =ᶠ[l] fun i => setToFun μ T hT (fs i)

Next tactic:
filter_upwards [hfsi] with i hi
/-- If `F i → f` in `L1`, then `setToFun μ T hT (F i) → setToFun μ T hT f`. -/
theorem tendsto_setToFun_of_L1 (hT : DominatedFinMeasAdditive μ T C) {ι} (f : α → E)
    (hfi : Integrable f μ) {fs : ι → α → E} {l : Filter ι} (hfsi : ∀ᶠ i in l, Integrable (fs i) μ)
    (hfs : Tendsto (fun i => ∫⁻ x, ‖fs i x - f x‖₊ ∂μ) l (𝓝 0)) :
    Tendsto (fun i => setToFun μ T hT (fs i)) l (𝓝 <| setToFun μ T hT f) := by
  classical
    let f_lp := hfi.toL1 f
    let F_lp i := if hFi : Integrable (fs i) μ then hFi.toL1 (fs i) else 0
    have tendsto_L1 : Tendsto F_lp l (𝓝 f_lp) := by
      rw [Lp.tendsto_Lp_iff_tendsto_ℒp']
      simp_rw [snorm_one_eq_lintegral_nnnorm, Pi.sub_apply]
      refine' (tendsto_congr' _).mp hfs
      filter_upwards [hfsi] with i hi
      refine' lintegral_congr_ae _
      filter_upwards [hi.coeFn_toL1, hfi.coeFn_toL1] with x hxi hxf
      simp_rw [dif_pos hi, hxi, hxf]
    suffices Tendsto (fun i => setToFun μ T hT (F_lp i)) l (𝓝 (setToFun μ T hT f)) by
      refine' (tendsto_congr' _).mp this
      filter_upwards [hfsi] with i hi
      

Proof state:
case h
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
f : α → E
hfi : Integrable f
fs : ι → α → E
l : Filter ι
hfsi : ∀ᶠ (i : ι) in l, Integrable (fs i)
hfs : Tendsto (fun i => ∫⁻ (x : α), ↑‖fs i x - f x‖₊ ∂μ) l (𝓝 0)
f_lp : ↥(Lp E 1) := Integrable.toL1 f hfi
F_lp : ι → ↥(Lp E 1) := fun i => if hFi : Integrable (fs i) then Integrable.toL1 (fs i) hFi else 0
tendsto_L1 : Tendsto F_lp l (𝓝 f_lp)
this : Tendsto (fun i => setToFun μ T hT ↑↑(F_lp i)) l (𝓝 (setToFun μ T hT f))
i : ι
hi : Integrable (fs i)
⊢ setToFun μ T hT ↑↑(F_lp i) = setToFun μ T hT (fs i)

Next tactic:
suffices h_ae_eq : F_lp i =ᵐ[μ] fs i
/-- If `F i → f` in `L1`, then `setToFun μ T hT (F i) → setToFun μ T hT f`. -/
theorem tendsto_setToFun_of_L1 (hT : DominatedFinMeasAdditive μ T C) {ι} (f : α → E)
    (hfi : Integrable f μ) {fs : ι → α → E} {l : Filter ι} (hfsi : ∀ᶠ i in l, Integrable (fs i) μ)
    (hfs : Tendsto (fun i => ∫⁻ x, ‖fs i x - f x‖₊ ∂μ) l (𝓝 0)) :
    Tendsto (fun i => setToFun μ T hT (fs i)) l (𝓝 <| setToFun μ T hT f) := by
  classical
    let f_lp := hfi.toL1 f
    let F_lp i := if hFi : Integrable (fs i) μ then hFi.toL1 (fs i) else 0
    have tendsto_L1 : Tendsto F_lp l (𝓝 f_lp) := by
      rw [Lp.tendsto_Lp_iff_tendsto_ℒp']
      simp_rw [snorm_one_eq_lintegral_nnnorm, Pi.sub_apply]
      refine' (tendsto_congr' _).mp hfs
      filter_upwards [hfsi] with i hi
      refine' lintegral_congr_ae _
      filter_upwards [hi.coeFn_toL1, hfi.coeFn_toL1] with x hxi hxf
      simp_rw [dif_pos hi, hxi, hxf]
    suffices Tendsto (fun i => setToFun μ T hT (F_lp i)) l (𝓝 (setToFun μ T hT f)) by
      refine' (tendsto_congr' _).mp this
      filter_upwards [hfsi] with i hi
      suffices h_ae_eq : F_lp i =ᵐ[μ] fs i; 

Proof state:
case h
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
f : α → E
hfi : Integrable f
fs : ι → α → E
l : Filter ι
hfsi : ∀ᶠ (i : ι) in l, Integrable (fs i)
hfs : Tendsto (fun i => ∫⁻ (x : α), ↑‖fs i x - f x‖₊ ∂μ) l (𝓝 0)
f_lp : ↥(Lp E 1) := Integrable.toL1 f hfi
F_lp : ι → ↥(Lp E 1) := fun i => if hFi : Integrable (fs i) then Integrable.toL1 (fs i) hFi else 0
tendsto_L1 : Tendsto F_lp l (𝓝 f_lp)
this : Tendsto (fun i => setToFun μ T hT ↑↑(F_lp i)) l (𝓝 (setToFun μ T hT f))
i : ι
hi : Integrable (fs i)
h_ae_eq : ↑↑(F_lp i) =ᵐ[μ] fs i
⊢ setToFun μ T hT ↑↑(F_lp i) = setToFun μ T hT (fs i)
case h_ae_eq
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
f : α → E
hfi : Integrable f
fs : ι → α → E
l : Filter ι
hfsi : ∀ᶠ (i : ι) in l, Integrable (fs i)
hfs : Tendsto (fun i => ∫⁻ (x : α), ↑‖fs i x - f x‖₊ ∂μ) l (𝓝 0)
f_lp : ↥(Lp E 1) := Integrable.toL1 f hfi
F_lp : ι → ↥(Lp E 1) := fun i => if hFi : Integrable (fs i) then Integrable.toL1 (fs i) hFi else 0
tendsto_L1 : Tendsto F_lp l (𝓝 f_lp)
this : Tendsto (fun i => setToFun μ T hT ↑↑(F_lp i)) l (𝓝 (setToFun μ T hT f))
i : ι
hi : Integrable (fs i)
⊢ ↑↑(F_lp i) =ᵐ[μ] fs i

Next tactic:
exact setToFun_congr_ae hT h_ae_eq
/-- If `F i → f` in `L1`, then `setToFun μ T hT (F i) → setToFun μ T hT f`. -/
theorem tendsto_setToFun_of_L1 (hT : DominatedFinMeasAdditive μ T C) {ι} (f : α → E)
    (hfi : Integrable f μ) {fs : ι → α → E} {l : Filter ι} (hfsi : ∀ᶠ i in l, Integrable (fs i) μ)
    (hfs : Tendsto (fun i => ∫⁻ x, ‖fs i x - f x‖₊ ∂μ) l (𝓝 0)) :
    Tendsto (fun i => setToFun μ T hT (fs i)) l (𝓝 <| setToFun μ T hT f) := by
  classical
    let f_lp := hfi.toL1 f
    let F_lp i := if hFi : Integrable (fs i) μ then hFi.toL1 (fs i) else 0
    have tendsto_L1 : Tendsto F_lp l (𝓝 f_lp) := by
      rw [Lp.tendsto_Lp_iff_tendsto_ℒp']
      simp_rw [snorm_one_eq_lintegral_nnnorm, Pi.sub_apply]
      refine' (tendsto_congr' _).mp hfs
      filter_upwards [hfsi] with i hi
      refine' lintegral_congr_ae _
      filter_upwards [hi.coeFn_toL1, hfi.coeFn_toL1] with x hxi hxf
      simp_rw [dif_pos hi, hxi, hxf]
    suffices Tendsto (fun i => setToFun μ T hT (F_lp i)) l (𝓝 (setToFun μ T hT f)) by
      refine' (tendsto_congr' _).mp this
      filter_upwards [hfsi] with i hi
      suffices h_ae_eq : F_lp i =ᵐ[μ] fs i; exact setToFun_congr_ae hT h_ae_eq
      

Proof state:
case h_ae_eq
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
f : α → E
hfi : Integrable f
fs : ι → α → E
l : Filter ι
hfsi : ∀ᶠ (i : ι) in l, Integrable (fs i)
hfs : Tendsto (fun i => ∫⁻ (x : α), ↑‖fs i x - f x‖₊ ∂μ) l (𝓝 0)
f_lp : ↥(Lp E 1) := Integrable.toL1 f hfi
F_lp : ι → ↥(Lp E 1) := fun i => if hFi : Integrable (fs i) then Integrable.toL1 (fs i) hFi else 0
tendsto_L1 : Tendsto F_lp l (𝓝 f_lp)
this : Tendsto (fun i => setToFun μ T hT ↑↑(F_lp i)) l (𝓝 (setToFun μ T hT f))
i : ι
hi : Integrable (fs i)
⊢ ↑↑(F_lp i) =ᵐ[μ] fs i

Next tactic:
simp_rw [dif_pos hi]