Theorem: MeasureTheory.AEFinStronglyMeasurable.ae_eq_of_forall_setIntegral_eq
File: Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean

Proof steps:
state_before: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ≥0∞
f g : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ
hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn g s μ
hfg_eq : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ
hf : AEFinStronglyMeasurable f μ
hg : AEFinStronglyMeasurable g μ
⊢ f =ᶠ[ae μ] g
state_after: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ≥0∞
f g : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ
hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn g s μ
hfg_eq : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ
hf : AEFinStronglyMeasurable f μ
hg : AEFinStronglyMeasurable g μ
⊢ f - g =ᶠ[ae μ] 0
tactic: rw [← sub_ae_eq_zero]
state_before: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ≥0∞
f g : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ
hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn g s μ
hfg_eq : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ
hf : AEFinStronglyMeasurable f μ
hg : AEFinStronglyMeasurable g μ
⊢ f - g =ᶠ[ae μ] 0
state_after: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ≥0∞
f g : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ
hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn g s μ
hfg_eq : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ
hf : AEFinStronglyMeasurable f μ
hg : AEFinStronglyMeasurable g μ
hfg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, (f - g) x ∂μ = 0
⊢ f - g =ᶠ[ae μ] 0
tactic: have hfg : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, (f - g) x ∂μ) = 0 := by
    intro s hs hμs
    rw [integral_sub' (hf_int_finite s hs hμs) (hg_int_finite s hs hμs),
      sub_eq_zero.mpr (hfg_eq s hs hμs)]
state_before: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ≥0∞
f g : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ
hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn g s μ
hfg_eq : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ
hf : AEFinStronglyMeasurable f μ
hg : AEFinStronglyMeasurable g μ
hfg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, (f - g) x ∂μ = 0
⊢ f - g =ᶠ[ae μ] 0
state_after: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ≥0∞
f g : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ
hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn g s μ
hfg_eq : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ
hf : AEFinStronglyMeasurable f μ
hg : AEFinStronglyMeasurable g μ
hfg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, (f - g) x ∂μ = 0
hfg_int : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn (f - g) s μ
⊢ f - g =ᶠ[ae μ] 0
tactic: have hfg_int : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn (f - g) s μ := fun s hs hμs =>
    (hf_int_finite s hs hμs).sub (hg_int_finite s hs hμs)
state_before: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ≥0∞
f g : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ
hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn g s μ
hfg_eq : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ
hf : AEFinStronglyMeasurable f μ
hg : AEFinStronglyMeasurable g μ
hfg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, (f - g) x ∂μ = 0
hfg_int : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn (f - g) s μ
⊢ f - g =ᶠ[ae μ] 0
state_after: no goals
tactic: exact (hf.sub hg).ae_eq_zero_of_forall_setIntegral_eq_zero hfg_int hfg