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']
      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
      simp_rw [dif_pos hi]
      

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)
⊢ ↑↑(Integrable.toL1 (fs i) hi) =ᵐ[μ] fs i

Next tactic:
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
      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
    

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 ↑↑(F_lp i)) l (𝓝 (setToFun μ T hT f))

Next tactic:
rw [setToFun_congr_ae hT hfi.coeFn_toL1.symm]
/-- 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
      simp_rw [dif_pos hi]
      exact hi.coeFn_toL1
    rw [setToFun_congr_ae hT hfi.coeFn_toL1.symm]
    

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 ↑↑(F_lp i)) l (𝓝 (setToFun μ T hT ↑↑(Integrable.toL1 f hfi)))

Next tactic:
exact ((continuous_setToFun hT).tendsto f_lp).comp tendsto_L1
theorem tendsto_setToFun_approxOn_of_measurable_of_range_subset
    (hT : DominatedFinMeasAdditive μ T C) [MeasurableSpace E] [BorelSpace E] {f : α → E}
    (fmeas : Measurable f) (hf : Integrable f μ) (s : Set E) [SeparableSpace s]
    (hs : range f ∪ {0} ⊆ s) :
    Tendsto (fun n => setToFun μ T hT (SimpleFunc.approxOn f fmeas s 0 (hs <| 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✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedAddCommGroup F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝³ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
f : α → E
fmeas : Measurable f
hf : Integrable f
s : Set E
inst✝ : SeparableSpace ↑s
hs : Set.range f ∪ {0} ⊆ s
n : ℕ
⊢ 0 ∈ Set.range f ∪ {0}

Next tactic:
simp
theorem tendsto_setToFun_approxOn_of_measurable_of_range_subset
    (hT : DominatedFinMeasAdditive μ T C) [MeasurableSpace E] [BorelSpace E] {f : α → E}
    (fmeas : Measurable f) (hf : Integrable f μ) (s : Set E) [SeparableSpace s]
    (hs : range f ∪ {0} ⊆ s) :
    Tendsto (fun n => setToFun μ T hT (SimpleFunc.approxOn f fmeas s 0 (hs <| by simp) n)) atTop
      (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✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedAddCommGroup F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝³ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
f : α → E
fmeas : Measurable f
hf : Integrable f
s : Set E
inst✝ : SeparableSpace ↑s
hs : Set.range f ∪ {0} ⊆ s
⊢ Tendsto (fun n => setToFun μ T hT ↑(SimpleFunc.approxOn f fmeas s 0 (_ : 0 ∈ s) n)) atTop (𝓝 (setToFun μ T hT f))

Next tactic:
refine tendsto_setToFun_approxOn_of_measurable hT hf fmeas ?_ _ (integrable_zero _ _ _)
theorem tendsto_setToFun_approxOn_of_measurable_of_range_subset
    (hT : DominatedFinMeasAdditive μ T C) [MeasurableSpace E] [BorelSpace E] {f : α → E}
    (fmeas : Measurable f) (hf : Integrable f μ) (s : Set E) [SeparableSpace s]
    (hs : range f ∪ {0} ⊆ s) :
    Tendsto (fun n => setToFun μ T hT (SimpleFunc.approxOn f fmeas s 0 (hs <| by simp) n)) atTop
      (𝓝 <| setToFun μ T hT f) := by
  refine tendsto_setToFun_approxOn_of_measurable hT hf fmeas ?_ _ (integrable_zero _ _ _)
  

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✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedAddCommGroup F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝³ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
f : α → E
fmeas : Measurable f
hf : Integrable f
s : Set E
inst✝ : SeparableSpace ↑s
hs : Set.range f ∪ {0} ⊆ s
⊢ ∀ᵐ (x : α) ∂μ, f x ∈ closure s

Next tactic:
exact eventually_of_forall fun x => subset_closure (hs (Set.mem_union_left _ (mem_range_self _)))
/-- Auxiliary lemma for `setToFun_congr_measure`: the function sending `f : α →₁[μ] G` to
`f : α →₁[μ'] G` is continuous when `μ' ≤ c' • μ` for `c' ≠ ∞`. -/
theorem continuous_L1_toL1 {μ' : Measure α} (c' : ℝ≥0∞) (hc' : c' ≠ ∞) (hμ'_le : μ' ≤ c' • μ) :
    Continuous fun f : α →₁[μ] G =>
      (Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f)).toL1 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
μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ⊤
hμ'_le : μ' ≤ c' • μ
⊢ Continuous fun f => Integrable.toL1 ↑↑f (_ : Integrable ↑↑f)

Next tactic:
by_cases hc'0 : c' = 0
/-- Auxiliary lemma for `setToFun_congr_measure`: the function sending `f : α →₁[μ] G` to
`f : α →₁[μ'] G` is continuous when `μ' ≤ c' • μ` for `c' ≠ ∞`. -/
theorem continuous_L1_toL1 {μ' : Measure α} (c' : ℝ≥0∞) (hc' : c' ≠ ∞) (hμ'_le : μ' ≤ c' • μ) :
    Continuous fun f : α →₁[μ] G =>
      (Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f)).toL1 f := by
  by_cases hc'0 : c' = 0
  · 

Proof state:
case pos
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
μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ⊤
hμ'_le : μ' ≤ c' • μ
hc'0 : c' = 0
⊢ Continuous fun f => Integrable.toL1 ↑↑f (_ : Integrable ↑↑f)

Next tactic:
have hμ'0 : μ' = 0 := by rw [← Measure.nonpos_iff_eq_zero']; refine' hμ'_le.trans _; simp [hc'0]
/-- Auxiliary lemma for `setToFun_congr_measure`: the function sending `f : α →₁[μ] G` to
`f : α →₁[μ'] G` is continuous when `μ' ≤ c' • μ` for `c' ≠ ∞`. -/
theorem continuous_L1_toL1 {μ' : Measure α} (c' : ℝ≥0∞) (hc' : c' ≠ ∞) (hμ'_le : μ' ≤ c' • μ) :
    Continuous fun f : α →₁[μ] G =>
      (Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f)).toL1 f := by
  by_cases hc'0 : c' = 0
  · have hμ'0 : μ' = 0 := 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
μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ⊤
hμ'_le : μ' ≤ c' • μ
hc'0 : c' = 0
⊢ μ' = 0

Next tactic:
rw [← Measure.nonpos_iff_eq_zero']
/-- Auxiliary lemma for `setToFun_congr_measure`: the function sending `f : α →₁[μ] G` to
`f : α →₁[μ'] G` is continuous when `μ' ≤ c' • μ` for `c' ≠ ∞`. -/
theorem continuous_L1_toL1 {μ' : Measure α} (c' : ℝ≥0∞) (hc' : c' ≠ ∞) (hμ'_le : μ' ≤ c' • μ) :
    Continuous fun f : α →₁[μ] G =>
      (Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f)).toL1 f := by
  by_cases hc'0 : c' = 0
  · have hμ'0 : μ' = 0 := by rw [← Measure.nonpos_iff_eq_zero']; 

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
μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ⊤
hμ'_le : μ' ≤ c' • μ
hc'0 : c' = 0
⊢ μ' ≤ 0

Next tactic:
refine' hμ'_le.trans _
/-- Auxiliary lemma for `setToFun_congr_measure`: the function sending `f : α →₁[μ] G` to
`f : α →₁[μ'] G` is continuous when `μ' ≤ c' • μ` for `c' ≠ ∞`. -/
theorem continuous_L1_toL1 {μ' : Measure α} (c' : ℝ≥0∞) (hc' : c' ≠ ∞) (hμ'_le : μ' ≤ c' • μ) :
    Continuous fun f : α →₁[μ] G =>
      (Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f)).toL1 f := by
  by_cases hc'0 : c' = 0
  · have hμ'0 : μ' = 0 := by rw [← Measure.nonpos_iff_eq_zero']; refine' hμ'_le.trans _; 

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
μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ⊤
hμ'_le : μ' ≤ c' • μ
hc'0 : c' = 0
⊢ c' • μ ≤ 0

Next tactic:
simp [hc'0]
/-- Auxiliary lemma for `setToFun_congr_measure`: the function sending `f : α →₁[μ] G` to
`f : α →₁[μ'] G` is continuous when `μ' ≤ c' • μ` for `c' ≠ ∞`. -/
theorem continuous_L1_toL1 {μ' : Measure α} (c' : ℝ≥0∞) (hc' : c' ≠ ∞) (hμ'_le : μ' ≤ c' • μ) :
    Continuous fun f : α →₁[μ] G =>
      (Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f)).toL1 f := by
  by_cases hc'0 : c' = 0
  · have hμ'0 : μ' = 0 := by rw [← Measure.nonpos_iff_eq_zero']; refine' hμ'_le.trans _; simp [hc'0]
    

Proof state:
case pos
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
μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ⊤
hμ'_le : μ' ≤ c' • μ
hc'0 : c' = 0
hμ'0 : μ' = 0
⊢ Continuous fun f => Integrable.toL1 ↑↑f (_ : Integrable ↑↑f)

Next tactic:
have h_im_zero :
      (fun f : α →₁[μ] G =>
          (Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f)).toL1 f) =
        0 := by
      ext1 f; ext1; simp_rw [hμ'0]; simp only [ae_zero, EventuallyEq, eventually_bot]
/-- Auxiliary lemma for `setToFun_congr_measure`: the function sending `f : α →₁[μ] G` to
`f : α →₁[μ'] G` is continuous when `μ' ≤ c' • μ` for `c' ≠ ∞`. -/
theorem continuous_L1_toL1 {μ' : Measure α} (c' : ℝ≥0∞) (hc' : c' ≠ ∞) (hμ'_le : μ' ≤ c' • μ) :
    Continuous fun f : α →₁[μ] G =>
      (Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f)).toL1 f := by
  by_cases hc'0 : c' = 0
  · have hμ'0 : μ' = 0 := by rw [← Measure.nonpos_iff_eq_zero']; refine' hμ'_le.trans _; simp [hc'0]
    have h_im_zero :
      (fun f : α →₁[μ] G =>
          (Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f)).toL1 f) =
        0 := 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
μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ⊤
hμ'_le : μ' ≤ c' • μ
hc'0 : c' = 0
hμ'0 : μ' = 0
⊢ (fun f => Integrable.toL1 ↑↑f (_ : Integrable ↑↑f)) = 0

Next tactic:
ext1 f
/-- Auxiliary lemma for `setToFun_congr_measure`: the function sending `f : α →₁[μ] G` to
`f : α →₁[μ'] G` is continuous when `μ' ≤ c' • μ` for `c' ≠ ∞`. -/
theorem continuous_L1_toL1 {μ' : Measure α} (c' : ℝ≥0∞) (hc' : c' ≠ ∞) (hμ'_le : μ' ≤ c' • μ) :
    Continuous fun f : α →₁[μ] G =>
      (Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f)).toL1 f := by
  by_cases hc'0 : c' = 0
  · have hμ'0 : μ' = 0 := by rw [← Measure.nonpos_iff_eq_zero']; refine' hμ'_le.trans _; simp [hc'0]
    have h_im_zero :
      (fun f : α →₁[μ] G =>
          (Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f)).toL1 f) =
        0 := by
      ext1 f; 

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
μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ⊤
hμ'_le : μ' ≤ c' • μ
hc'0 : c' = 0
hμ'0 : μ' = 0
f : ↥(Lp G 1)
⊢ Integrable.toL1 ↑↑f (_ : Integrable ↑↑f) = OfNat.ofNat 0 f

Next tactic:
ext1
/-- Auxiliary lemma for `setToFun_congr_measure`: the function sending `f : α →₁[μ] G` to
`f : α →₁[μ'] G` is continuous when `μ' ≤ c' • μ` for `c' ≠ ∞`. -/
theorem continuous_L1_toL1 {μ' : Measure α} (c' : ℝ≥0∞) (hc' : c' ≠ ∞) (hμ'_le : μ' ≤ c' • μ) :
    Continuous fun f : α →₁[μ] G =>
      (Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f)).toL1 f := by
  by_cases hc'0 : c' = 0
  · have hμ'0 : μ' = 0 := by rw [← Measure.nonpos_iff_eq_zero']; refine' hμ'_le.trans _; simp [hc'0]
    have h_im_zero :
      (fun f : α →₁[μ] G =>
          (Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f)).toL1 f) =
        0 := by
      ext1 f; ext1; 

Proof state:
case h.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
μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ⊤
hμ'_le : μ' ≤ c' • μ
hc'0 : c' = 0
hμ'0 : μ' = 0
f : ↥(Lp G 1)
⊢ ↑↑(Integrable.toL1 ↑↑f (_ : Integrable ↑↑f)) =ᵐ[μ'] ↑↑(OfNat.ofNat 0 f)

Next tactic:
simp_rw [hμ'0]
/-- Auxiliary lemma for `setToFun_congr_measure`: the function sending `f : α →₁[μ] G` to
`f : α →₁[μ'] G` is continuous when `μ' ≤ c' • μ` for `c' ≠ ∞`. -/
theorem continuous_L1_toL1 {μ' : Measure α} (c' : ℝ≥0∞) (hc' : c' ≠ ∞) (hμ'_le : μ' ≤ c' • μ) :
    Continuous fun f : α →₁[μ] G =>
      (Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f)).toL1 f := by
  by_cases hc'0 : c' = 0
  · have hμ'0 : μ' = 0 := by rw [← Measure.nonpos_iff_eq_zero']; refine' hμ'_le.trans _; simp [hc'0]
    have h_im_zero :
      (fun f : α →₁[μ] G =>
          (Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f)).toL1 f) =
        0 := by
      ext1 f; ext1; simp_rw [hμ'0]; 

Proof state:
case h.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
μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ⊤
hμ'_le : μ' ≤ c' • μ
hc'0 : c' = 0
hμ'0 : μ' = 0
f : ↥(Lp G 1)
⊢ ↑↑(Integrable.toL1 ↑↑f (_ : Integrable ↑↑f)) =ᵐ[0] ↑↑(OfNat.ofNat 0 f)

Next tactic:
simp only [ae_zero, EventuallyEq, eventually_bot]
/-- Auxiliary lemma for `setToFun_congr_measure`: the function sending `f : α →₁[μ] G` to
`f : α →₁[μ'] G` is continuous when `μ' ≤ c' • μ` for `c' ≠ ∞`. -/
theorem continuous_L1_toL1 {μ' : Measure α} (c' : ℝ≥0∞) (hc' : c' ≠ ∞) (hμ'_le : μ' ≤ c' • μ) :
    Continuous fun f : α →₁[μ] G =>
      (Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f)).toL1 f := by
  by_cases hc'0 : c' = 0
  · have hμ'0 : μ' = 0 := by rw [← Measure.nonpos_iff_eq_zero']; refine' hμ'_le.trans _; simp [hc'0]
    have h_im_zero :
      (fun f : α →₁[μ] G =>
          (Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f)).toL1 f) =
        0 := by
      ext1 f; ext1; simp_rw [hμ'0]; simp only [ae_zero, EventuallyEq, eventually_bot]
    

Proof state:
case pos
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
μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ⊤
hμ'_le : μ' ≤ c' • μ
hc'0 : c' = 0
hμ'0 : μ' = 0
h_im_zero : (fun f => Integrable.toL1 ↑↑f (_ : Integrable ↑↑f)) = 0
⊢ Continuous fun f => Integrable.toL1 ↑↑f (_ : Integrable ↑↑f)

Next tactic:
rw [h_im_zero]
/-- Auxiliary lemma for `setToFun_congr_measure`: the function sending `f : α →₁[μ] G` to
`f : α →₁[μ'] G` is continuous when `μ' ≤ c' • μ` for `c' ≠ ∞`. -/
theorem continuous_L1_toL1 {μ' : Measure α} (c' : ℝ≥0∞) (hc' : c' ≠ ∞) (hμ'_le : μ' ≤ c' • μ) :
    Continuous fun f : α →₁[μ] G =>
      (Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f)).toL1 f := by
  by_cases hc'0 : c' = 0
  · have hμ'0 : μ' = 0 := by rw [← Measure.nonpos_iff_eq_zero']; refine' hμ'_le.trans _; simp [hc'0]
    have h_im_zero :
      (fun f : α →₁[μ] G =>
          (Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f)).toL1 f) =
        0 := by
      ext1 f; ext1; simp_rw [hμ'0]; simp only [ae_zero, EventuallyEq, eventually_bot]
    rw [h_im_zero]
    

Proof state:
case pos
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
μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ⊤
hμ'_le : μ' ≤ c' • μ
hc'0 : c' = 0
hμ'0 : μ' = 0
h_im_zero : (fun f => Integrable.toL1 ↑↑f (_ : Integrable ↑↑f)) = 0
⊢ Continuous 0

Next tactic:
exact continuous_zero