l3lab/ntp-mathlib huggingface dataset:
theorem lpMeas.ae_eq_zero_of_forall_set_integral_eq_zero (hm : m ≤ m0) (f : lpMeas E' 𝕜 m p μ)
    (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
    -- Porting note: needed to add explicit casts in the next two hypotheses
    (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn (f : Lp E' p μ) s μ)
    (hf_zero : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, (f : Lp E' p μ) x ∂μ = 0) :
    f =ᵐ[μ] (0 : α → E') := by
  obtain ⟨g, hg_sm, hfg⟩ := lpMeas.ae_fin_strongly_measurable' hm f hp_ne_zero hp_ne_top
  refine' hfg.trans _
  -- Porting note: added
  unfold Filter.EventuallyEq at hfg
  refine' ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim hm _ _ hg_sm
  · intro s hs hμs
    have hfg_restrict : f =ᵐ[μ.restrict s] g := ae_restrict_of_ae hfg
    rw [IntegrableOn, integrable_congr hfg_restrict.symm]
    exact hf_int_finite s hs hμs
  · 

Proof state:
case intro.intro.refine'_2
α : Type u_1
E' : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
m m0 : MeasurableSpace α
μ : Measure α
inst✝⁸ : IsROrC 𝕜
inst✝⁷ : NormedAddCommGroup E'
inst✝⁶ : InnerProductSpace 𝕜 E'
inst✝⁵ : CompleteSpace E'
inst✝⁴ : NormedSpace ℝ E'
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
hm : m ≤ m0
f : ↥(lpMeas E' 𝕜 m p μ)
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ⊤
hf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn (↑↑↑f) s
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∫ (x : α) in s, ↑↑↑f x ∂μ = 0
g : α → E'
hg_sm : FinStronglyMeasurable g (Measure.trim μ hm)
hfg : ∀ᵐ (x : α) ∂μ, ↑↑↑f x = g x
⊢ ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∫ (x : α) in s, g x ∂μ = 0

Next tactic:
intro s hs hμs
theorem lpMeas.ae_eq_zero_of_forall_set_integral_eq_zero (hm : m ≤ m0) (f : lpMeas E' 𝕜 m p μ)
    (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
    -- Porting note: needed to add explicit casts in the next two hypotheses
    (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn (f : Lp E' p μ) s μ)
    (hf_zero : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, (f : Lp E' p μ) x ∂μ = 0) :
    f =ᵐ[μ] (0 : α → E') := by
  obtain ⟨g, hg_sm, hfg⟩ := lpMeas.ae_fin_strongly_measurable' hm f hp_ne_zero hp_ne_top
  refine' hfg.trans _
  -- Porting note: added
  unfold Filter.EventuallyEq at hfg
  refine' ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim hm _ _ hg_sm
  · intro s hs hμs
    have hfg_restrict : f =ᵐ[μ.restrict s] g := ae_restrict_of_ae hfg
    rw [IntegrableOn, integrable_congr hfg_restrict.symm]
    exact hf_int_finite s hs hμs
  · intro s hs hμs
    

Proof state:
case intro.intro.refine'_2
α : Type u_1
E' : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
m m0 : MeasurableSpace α
μ : Measure α
inst✝⁸ : IsROrC 𝕜
inst✝⁷ : NormedAddCommGroup E'
inst✝⁶ : InnerProductSpace 𝕜 E'
inst✝⁵ : CompleteSpace E'
inst✝⁴ : NormedSpace ℝ E'
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
hm : m ≤ m0
f : ↥(lpMeas E' 𝕜 m p μ)
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ⊤
hf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn (↑↑↑f) s
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∫ (x : α) in s, ↑↑↑f x ∂μ = 0
g : α → E'
hg_sm : FinStronglyMeasurable g (Measure.trim μ hm)
hfg : ∀ᵐ (x : α) ∂μ, ↑↑↑f x = g x
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s < ⊤
⊢ ∫ (x : α) in s, g x ∂μ = 0

Next tactic:
have hfg_restrict : f =ᵐ[μ.restrict s] g := ae_restrict_of_ae hfg
theorem lpMeas.ae_eq_zero_of_forall_set_integral_eq_zero (hm : m ≤ m0) (f : lpMeas E' 𝕜 m p μ)
    (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
    -- Porting note: needed to add explicit casts in the next two hypotheses
    (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn (f : Lp E' p μ) s μ)
    (hf_zero : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, (f : Lp E' p μ) x ∂μ = 0) :
    f =ᵐ[μ] (0 : α → E') := by
  obtain ⟨g, hg_sm, hfg⟩ := lpMeas.ae_fin_strongly_measurable' hm f hp_ne_zero hp_ne_top
  refine' hfg.trans _
  -- Porting note: added
  unfold Filter.EventuallyEq at hfg
  refine' ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim hm _ _ hg_sm
  · intro s hs hμs
    have hfg_restrict : f =ᵐ[μ.restrict s] g := ae_restrict_of_ae hfg
    rw [IntegrableOn, integrable_congr hfg_restrict.symm]
    exact hf_int_finite s hs hμs
  · intro s hs hμs
    have hfg_restrict : f =ᵐ[μ.restrict s] g := ae_restrict_of_ae hfg
    

Proof state:
case intro.intro.refine'_2
α : Type u_1
E' : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
m m0 : MeasurableSpace α
μ : Measure α
inst✝⁸ : IsROrC 𝕜
inst✝⁷ : NormedAddCommGroup E'
inst✝⁶ : InnerProductSpace 𝕜 E'
inst✝⁵ : CompleteSpace E'
inst✝⁴ : NormedSpace ℝ E'
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
hm : m ≤ m0
f : ↥(lpMeas E' 𝕜 m p μ)
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ⊤
hf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn (↑↑↑f) s
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∫ (x : α) in s, ↑↑↑f x ∂μ = 0
g : α → E'
hg_sm : FinStronglyMeasurable g (Measure.trim μ hm)
hfg : ∀ᵐ (x : α) ∂μ, ↑↑↑f x = g x
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s < ⊤
hfg_restrict : ↑↑↑f =ᵐ[Measure.restrict μ s] g
⊢ ∫ (x : α) in s, g x ∂μ = 0

Next tactic:
rw [integral_congr_ae hfg_restrict.symm]
theorem lpMeas.ae_eq_zero_of_forall_set_integral_eq_zero (hm : m ≤ m0) (f : lpMeas E' 𝕜 m p μ)
    (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
    -- Porting note: needed to add explicit casts in the next two hypotheses
    (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn (f : Lp E' p μ) s μ)
    (hf_zero : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, (f : Lp E' p μ) x ∂μ = 0) :
    f =ᵐ[μ] (0 : α → E') := by
  obtain ⟨g, hg_sm, hfg⟩ := lpMeas.ae_fin_strongly_measurable' hm f hp_ne_zero hp_ne_top
  refine' hfg.trans _
  -- Porting note: added
  unfold Filter.EventuallyEq at hfg
  refine' ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim hm _ _ hg_sm
  · intro s hs hμs
    have hfg_restrict : f =ᵐ[μ.restrict s] g := ae_restrict_of_ae hfg
    rw [IntegrableOn, integrable_congr hfg_restrict.symm]
    exact hf_int_finite s hs hμs
  · intro s hs hμs
    have hfg_restrict : f =ᵐ[μ.restrict s] g := ae_restrict_of_ae hfg
    rw [integral_congr_ae hfg_restrict.symm]
    

Proof state:
case intro.intro.refine'_2
α : Type u_1
E' : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
m m0 : MeasurableSpace α
μ : Measure α
inst✝⁸ : IsROrC 𝕜
inst✝⁷ : NormedAddCommGroup E'
inst✝⁶ : InnerProductSpace 𝕜 E'
inst✝⁵ : CompleteSpace E'
inst✝⁴ : NormedSpace ℝ E'
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
hm : m ≤ m0
f : ↥(lpMeas E' 𝕜 m p μ)
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ⊤
hf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn (↑↑↑f) s
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∫ (x : α) in s, ↑↑↑f x ∂μ = 0
g : α → E'
hg_sm : FinStronglyMeasurable g (Measure.trim μ hm)
hfg : ∀ᵐ (x : α) ∂μ, ↑↑↑f x = g x
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s < ⊤
hfg_restrict : ↑↑↑f =ᵐ[Measure.restrict μ s] g
⊢ ∫ (a : α) in s, ↑↑↑f a ∂μ = 0

Next tactic:
exact hf_zero s hs hμs
theorem Lp.ae_eq_zero_of_forall_set_integral_eq_zero' (hm : m ≤ m0) (f : Lp E' p μ)
    (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
    (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn f s μ)
    (hf_zero : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, f x ∂μ = 0)
    (hf_meas : AEStronglyMeasurable' m f μ) : f =ᵐ[μ] 0 := by
  

Proof state:
α : Type u_1
E' : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
m m0 : MeasurableSpace α
μ : Measure α
inst✝⁸ : IsROrC 𝕜
inst✝⁷ : NormedAddCommGroup E'
inst✝⁶ : InnerProductSpace 𝕜 E'
inst✝⁵ : CompleteSpace E'
inst✝⁴ : NormedSpace ℝ E'
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
hm : m ≤ m0
f : ↥(Lp E' p)
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ⊤
hf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn (↑↑f) s
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∫ (x : α) in s, ↑↑f x ∂μ = 0
hf_meas : AEStronglyMeasurable' m (↑↑f) μ
⊢ ↑↑f =ᵐ[μ] 0

Next tactic:
let f_meas : lpMeas E' 𝕜 m p μ := ⟨f, hf_meas⟩
theorem Lp.ae_eq_zero_of_forall_set_integral_eq_zero' (hm : m ≤ m0) (f : Lp E' p μ)
    (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
    (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn f s μ)
    (hf_zero : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, f x ∂μ = 0)
    (hf_meas : AEStronglyMeasurable' m f μ) : f =ᵐ[μ] 0 := by
  let f_meas : lpMeas E' 𝕜 m p μ := ⟨f, hf_meas⟩
  -- Porting note: `simp only` does not call `rfl` to try to close the goal. See https://github.com/leanprover-community/mathlib4/issues/5025
  

Proof state:
α : Type u_1
E' : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
m m0 : MeasurableSpace α
μ : Measure α
inst✝⁸ : IsROrC 𝕜
inst✝⁷ : NormedAddCommGroup E'
inst✝⁶ : InnerProductSpace 𝕜 E'
inst✝⁵ : CompleteSpace E'
inst✝⁴ : NormedSpace ℝ E'
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
hm : m ≤ m0
f : ↥(Lp E' p)
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ⊤
hf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn (↑↑f) s
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∫ (x : α) in s, ↑↑f x ∂μ = 0
hf_meas : AEStronglyMeasurable' m (↑↑f) μ
f_meas : ↥(lpMeas E' 𝕜 m p μ) := { val := f, property := hf_meas }
⊢ ↑↑f =ᵐ[μ] 0

Next tactic:
have hf_f_meas : f =ᵐ[μ] f_meas := by simp only [Subtype.coe_mk]; rfl
theorem Lp.ae_eq_zero_of_forall_set_integral_eq_zero' (hm : m ≤ m0) (f : Lp E' p μ)
    (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
    (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn f s μ)
    (hf_zero : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, f x ∂μ = 0)
    (hf_meas : AEStronglyMeasurable' m f μ) : f =ᵐ[μ] 0 := by
  let f_meas : lpMeas E' 𝕜 m p μ := ⟨f, hf_meas⟩
  -- Porting note: `simp only` does not call `rfl` to try to close the goal. See https://github.com/leanprover-community/mathlib4/issues/5025
  have hf_f_meas : f =ᵐ[μ] f_meas := by 

Proof state:
α : Type u_1
E' : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
m m0 : MeasurableSpace α
μ : Measure α
inst✝⁸ : IsROrC 𝕜
inst✝⁷ : NormedAddCommGroup E'
inst✝⁶ : InnerProductSpace 𝕜 E'
inst✝⁵ : CompleteSpace E'
inst✝⁴ : NormedSpace ℝ E'
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
hm : m ≤ m0
f : ↥(Lp E' p)
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ⊤
hf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn (↑↑f) s
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∫ (x : α) in s, ↑↑f x ∂μ = 0
hf_meas : AEStronglyMeasurable' m (↑↑f) μ
f_meas : ↥(lpMeas E' 𝕜 m p μ) := { val := f, property := hf_meas }
⊢ ↑↑f =ᵐ[μ] ↑↑↑f_meas

Next tactic:
simp only [Subtype.coe_mk]
theorem Lp.ae_eq_zero_of_forall_set_integral_eq_zero' (hm : m ≤ m0) (f : Lp E' p μ)
    (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
    (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn f s μ)
    (hf_zero : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, f x ∂μ = 0)
    (hf_meas : AEStronglyMeasurable' m f μ) : f =ᵐ[μ] 0 := by
  let f_meas : lpMeas E' 𝕜 m p μ := ⟨f, hf_meas⟩
  -- Porting note: `simp only` does not call `rfl` to try to close the goal. See https://github.com/leanprover-community/mathlib4/issues/5025
  have hf_f_meas : f =ᵐ[μ] f_meas := by simp only [Subtype.coe_mk]; 

Proof state:
α : Type u_1
E' : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
m m0 : MeasurableSpace α
μ : Measure α
inst✝⁸ : IsROrC 𝕜
inst✝⁷ : NormedAddCommGroup E'
inst✝⁶ : InnerProductSpace 𝕜 E'
inst✝⁵ : CompleteSpace E'
inst✝⁴ : NormedSpace ℝ E'
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
hm : m ≤ m0
f : ↥(Lp E' p)
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ⊤
hf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn (↑↑f) s
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∫ (x : α) in s, ↑↑f x ∂μ = 0
hf_meas : AEStronglyMeasurable' m (↑↑f) μ
f_meas : ↥(lpMeas E' 𝕜 m p μ) := { val := f, property := hf_meas }
⊢ ↑↑f =ᵐ[μ] ↑↑f

Next tactic:
rfl
theorem Lp.ae_eq_zero_of_forall_set_integral_eq_zero' (hm : m ≤ m0) (f : Lp E' p μ)
    (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
    (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn f s μ)
    (hf_zero : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, f x ∂μ = 0)
    (hf_meas : AEStronglyMeasurable' m f μ) : f =ᵐ[μ] 0 := by
  let f_meas : lpMeas E' 𝕜 m p μ := ⟨f, hf_meas⟩
  -- Porting note: `simp only` does not call `rfl` to try to close the goal. See https://github.com/leanprover-community/mathlib4/issues/5025
  have hf_f_meas : f =ᵐ[μ] f_meas := by simp only [Subtype.coe_mk]; rfl
  

Proof state:
α : Type u_1
E' : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
m m0 : MeasurableSpace α
μ : Measure α
inst✝⁸ : IsROrC 𝕜
inst✝⁷ : NormedAddCommGroup E'
inst✝⁶ : InnerProductSpace 𝕜 E'
inst✝⁵ : CompleteSpace E'
inst✝⁴ : NormedSpace ℝ E'
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
hm : m ≤ m0
f : ↥(Lp E' p)
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ⊤
hf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn (↑↑f) s
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∫ (x : α) in s, ↑↑f x ∂μ = 0
hf_meas : AEStronglyMeasurable' m (↑↑f) μ
f_meas : ↥(lpMeas E' 𝕜 m p μ) := { val := f, property := hf_meas }
hf_f_meas : ↑↑f =ᵐ[μ] ↑↑↑f_meas
⊢ ↑↑f =ᵐ[μ] 0

Next tactic:
refine' hf_f_meas.trans _
theorem Lp.ae_eq_zero_of_forall_set_integral_eq_zero' (hm : m ≤ m0) (f : Lp E' p μ)
    (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
    (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn f s μ)
    (hf_zero : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, f x ∂μ = 0)
    (hf_meas : AEStronglyMeasurable' m f μ) : f =ᵐ[μ] 0 := by
  let f_meas : lpMeas E' 𝕜 m p μ := ⟨f, hf_meas⟩
  -- Porting note: `simp only` does not call `rfl` to try to close the goal. See https://github.com/leanprover-community/mathlib4/issues/5025
  have hf_f_meas : f =ᵐ[μ] f_meas := by simp only [Subtype.coe_mk]; rfl
  refine' hf_f_meas.trans _
  

Proof state:
α : Type u_1
E' : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
m m0 : MeasurableSpace α
μ : Measure α
inst✝⁸ : IsROrC 𝕜
inst✝⁷ : NormedAddCommGroup E'
inst✝⁶ : InnerProductSpace 𝕜 E'
inst✝⁵ : CompleteSpace E'
inst✝⁴ : NormedSpace ℝ E'
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
hm : m ≤ m0
f : ↥(Lp E' p)
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ⊤
hf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn (↑↑f) s
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∫ (x : α) in s, ↑↑f x ∂μ = 0
hf_meas : AEStronglyMeasurable' m (↑↑f) μ
f_meas : ↥(lpMeas E' 𝕜 m p μ) := { val := f, property := hf_meas }
hf_f_meas : ↑↑f =ᵐ[μ] ↑↑↑f_meas
⊢ ↑↑↑f_meas =ᵐ[μ] 0

Next tactic:
refine' lpMeas.ae_eq_zero_of_forall_set_integral_eq_zero hm f_meas hp_ne_zero hp_ne_top _ _
theorem Lp.ae_eq_zero_of_forall_set_integral_eq_zero' (hm : m ≤ m0) (f : Lp E' p μ)
    (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
    (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn f s μ)
    (hf_zero : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, f x ∂μ = 0)
    (hf_meas : AEStronglyMeasurable' m f μ) : f =ᵐ[μ] 0 := by
  let f_meas : lpMeas E' 𝕜 m p μ := ⟨f, hf_meas⟩
  -- Porting note: `simp only` does not call `rfl` to try to close the goal. See https://github.com/leanprover-community/mathlib4/issues/5025
  have hf_f_meas : f =ᵐ[μ] f_meas := by simp only [Subtype.coe_mk]; rfl
  refine' hf_f_meas.trans _
  refine' lpMeas.ae_eq_zero_of_forall_set_integral_eq_zero hm f_meas hp_ne_zero hp_ne_top _ _
  · 

Proof state:
case refine'_1
α : Type u_1
E' : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
m m0 : MeasurableSpace α
μ : Measure α
inst✝⁸ : IsROrC 𝕜
inst✝⁷ : NormedAddCommGroup E'
inst✝⁶ : InnerProductSpace 𝕜 E'
inst✝⁵ : CompleteSpace E'
inst✝⁴ : NormedSpace ℝ E'
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
hm : m ≤ m0
f : ↥(Lp E' p)
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ⊤
hf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn (↑↑f) s
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∫ (x : α) in s, ↑↑f x ∂μ = 0
hf_meas : AEStronglyMeasurable' m (↑↑f) μ
f_meas : ↥(lpMeas E' 𝕜 m p μ) := { val := f, property := hf_meas }
hf_f_meas : ↑↑f =ᵐ[μ] ↑↑↑f_meas
⊢ ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn (↑↑↑f_meas) s

Next tactic:
intro s hs hμs
theorem Lp.ae_eq_zero_of_forall_set_integral_eq_zero' (hm : m ≤ m0) (f : Lp E' p μ)
    (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
    (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn f s μ)
    (hf_zero : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, f x ∂μ = 0)
    (hf_meas : AEStronglyMeasurable' m f μ) : f =ᵐ[μ] 0 := by
  let f_meas : lpMeas E' 𝕜 m p μ := ⟨f, hf_meas⟩
  -- Porting note: `simp only` does not call `rfl` to try to close the goal. See https://github.com/leanprover-community/mathlib4/issues/5025
  have hf_f_meas : f =ᵐ[μ] f_meas := by simp only [Subtype.coe_mk]; rfl
  refine' hf_f_meas.trans _
  refine' lpMeas.ae_eq_zero_of_forall_set_integral_eq_zero hm f_meas hp_ne_zero hp_ne_top _ _
  · intro s hs hμs
    

Proof state:
case refine'_1
α : Type u_1
E' : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
m m0 : MeasurableSpace α
μ : Measure α
inst✝⁸ : IsROrC 𝕜
inst✝⁷ : NormedAddCommGroup E'
inst✝⁶ : InnerProductSpace 𝕜 E'
inst✝⁵ : CompleteSpace E'
inst✝⁴ : NormedSpace ℝ E'
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
hm : m ≤ m0
f : ↥(Lp E' p)
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ⊤
hf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn (↑↑f) s
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∫ (x : α) in s, ↑↑f x ∂μ = 0
hf_meas : AEStronglyMeasurable' m (↑↑f) μ
f_meas : ↥(lpMeas E' 𝕜 m p μ) := { val := f, property := hf_meas }
hf_f_meas : ↑↑f =ᵐ[μ] ↑↑↑f_meas
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s < ⊤
⊢ IntegrableOn (↑↑↑f_meas) s

Next tactic:
have hfg_restrict : f =ᵐ[μ.restrict s] f_meas := ae_restrict_of_ae hf_f_meas
theorem Lp.ae_eq_zero_of_forall_set_integral_eq_zero' (hm : m ≤ m0) (f : Lp E' p μ)
    (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
    (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn f s μ)
    (hf_zero : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, f x ∂μ = 0)
    (hf_meas : AEStronglyMeasurable' m f μ) : f =ᵐ[μ] 0 := by
  let f_meas : lpMeas E' 𝕜 m p μ := ⟨f, hf_meas⟩
  -- Porting note: `simp only` does not call `rfl` to try to close the goal. See https://github.com/leanprover-community/mathlib4/issues/5025
  have hf_f_meas : f =ᵐ[μ] f_meas := by simp only [Subtype.coe_mk]; rfl
  refine' hf_f_meas.trans _
  refine' lpMeas.ae_eq_zero_of_forall_set_integral_eq_zero hm f_meas hp_ne_zero hp_ne_top _ _
  · intro s hs hμs
    have hfg_restrict : f =ᵐ[μ.restrict s] f_meas := ae_restrict_of_ae hf_f_meas
    

Proof state:
case refine'_1
α : Type u_1
E' : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
m m0 : MeasurableSpace α
μ : Measure α
inst✝⁸ : IsROrC 𝕜
inst✝⁷ : NormedAddCommGroup E'
inst✝⁶ : InnerProductSpace 𝕜 E'
inst✝⁵ : CompleteSpace E'
inst✝⁴ : NormedSpace ℝ E'
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
hm : m ≤ m0
f : ↥(Lp E' p)
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ⊤
hf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn (↑↑f) s
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∫ (x : α) in s, ↑↑f x ∂μ = 0
hf_meas : AEStronglyMeasurable' m (↑↑f) μ
f_meas : ↥(lpMeas E' 𝕜 m p μ) := { val := f, property := hf_meas }
hf_f_meas : ↑↑f =ᵐ[μ] ↑↑↑f_meas
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s < ⊤
hfg_restrict : ↑↑f =ᵐ[Measure.restrict μ s] ↑↑↑f_meas
⊢ IntegrableOn (↑↑↑f_meas) s

Next tactic:
rw [IntegrableOn, integrable_congr hfg_restrict.symm]
theorem Lp.ae_eq_zero_of_forall_set_integral_eq_zero' (hm : m ≤ m0) (f : Lp E' p μ)
    (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
    (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn f s μ)
    (hf_zero : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, f x ∂μ = 0)
    (hf_meas : AEStronglyMeasurable' m f μ) : f =ᵐ[μ] 0 := by
  let f_meas : lpMeas E' 𝕜 m p μ := ⟨f, hf_meas⟩
  -- Porting note: `simp only` does not call `rfl` to try to close the goal. See https://github.com/leanprover-community/mathlib4/issues/5025
  have hf_f_meas : f =ᵐ[μ] f_meas := by simp only [Subtype.coe_mk]; rfl
  refine' hf_f_meas.trans _
  refine' lpMeas.ae_eq_zero_of_forall_set_integral_eq_zero hm f_meas hp_ne_zero hp_ne_top _ _
  · intro s hs hμs
    have hfg_restrict : f =ᵐ[μ.restrict s] f_meas := ae_restrict_of_ae hf_f_meas
    rw [IntegrableOn, integrable_congr hfg_restrict.symm]
    

Proof state:
case refine'_1
α : Type u_1
E' : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
m m0 : MeasurableSpace α
μ : Measure α
inst✝⁸ : IsROrC 𝕜
inst✝⁷ : NormedAddCommGroup E'
inst✝⁶ : InnerProductSpace 𝕜 E'
inst✝⁵ : CompleteSpace E'
inst✝⁴ : NormedSpace ℝ E'
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
hm : m ≤ m0
f : ↥(Lp E' p)
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ⊤
hf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn (↑↑f) s
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∫ (x : α) in s, ↑↑f x ∂μ = 0
hf_meas : AEStronglyMeasurable' m (↑↑f) μ
f_meas : ↥(lpMeas E' 𝕜 m p μ) := { val := f, property := hf_meas }
hf_f_meas : ↑↑f =ᵐ[μ] ↑↑↑f_meas
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s < ⊤
hfg_restrict : ↑↑f =ᵐ[Measure.restrict μ s] ↑↑↑f_meas
⊢ Integrable ↑↑f

Next tactic:
exact hf_int_finite s hs hμs
theorem Lp.ae_eq_zero_of_forall_set_integral_eq_zero' (hm : m ≤ m0) (f : Lp E' p μ)
    (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
    (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn f s μ)
    (hf_zero : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, f x ∂μ = 0)
    (hf_meas : AEStronglyMeasurable' m f μ) : f =ᵐ[μ] 0 := by
  let f_meas : lpMeas E' 𝕜 m p μ := ⟨f, hf_meas⟩
  -- Porting note: `simp only` does not call `rfl` to try to close the goal. See https://github.com/leanprover-community/mathlib4/issues/5025
  have hf_f_meas : f =ᵐ[μ] f_meas := by simp only [Subtype.coe_mk]; rfl
  refine' hf_f_meas.trans _
  refine' lpMeas.ae_eq_zero_of_forall_set_integral_eq_zero hm f_meas hp_ne_zero hp_ne_top _ _
  · intro s hs hμs
    have hfg_restrict : f =ᵐ[μ.restrict s] f_meas := ae_restrict_of_ae hf_f_meas
    rw [IntegrableOn, integrable_congr hfg_restrict.symm]
    exact hf_int_finite s hs hμs
  · 

Proof state:
case refine'_2
α : Type u_1
E' : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
m m0 : MeasurableSpace α
μ : Measure α
inst✝⁸ : IsROrC 𝕜
inst✝⁷ : NormedAddCommGroup E'
inst✝⁶ : InnerProductSpace 𝕜 E'
inst✝⁵ : CompleteSpace E'
inst✝⁴ : NormedSpace ℝ E'
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
hm : m ≤ m0
f : ↥(Lp E' p)
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ⊤
hf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn (↑↑f) s
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∫ (x : α) in s, ↑↑f x ∂μ = 0
hf_meas : AEStronglyMeasurable' m (↑↑f) μ
f_meas : ↥(lpMeas E' 𝕜 m p μ) := { val := f, property := hf_meas }
hf_f_meas : ↑↑f =ᵐ[μ] ↑↑↑f_meas
⊢ ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∫ (x : α) in s, ↑↑↑f_meas x ∂μ = 0

Next tactic:
intro s hs hμs