Theorem: MeasureTheory.ae_eq_restrict_of_forall_set_integral_eq
File: Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean

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
hg_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn g s
hfg_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, f x ∂μ) = ∫ (x : α) in s, g x ∂μ
t : Set α
ht : MeasurableSet t
hμt : ↑↑μ t ≠ ⊤
⊢ f =ᵐ[Measure.restrict μ t] g

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)]
  exact sub_eq_zero.mpr (hfg_zero 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
hg_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn g s
hfg_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, f x ∂μ) = ∫ (x : α) in s, g x ∂μ
t : Set α
ht : MeasurableSet t
hμt : ↑↑μ t ≠ ⊤
⊢ f - g =ᵐ[Measure.restrict μ t] 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
hg_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn g s
hfg_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, f x ∂μ) = ∫ (x : α) in s, g x ∂μ
t : Set α
ht : MeasurableSet t
hμt : ↑↑μ t ≠ ⊤
hfg' : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, (f - g) x ∂μ) = 0
⊢ f - g =ᵐ[Measure.restrict μ t] 0

tactic: exact ae_eq_zero_restrict_of_forall_set_integral_eq_zero hfg_int hfg' ht hμt
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
hg_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn g s
hfg_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, f x ∂μ) = ∫ (x : α) in s, g x ∂μ
t : Set α
ht : MeasurableSet t
hμt : ↑↑μ t ≠ ⊤
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
⊢ f - g =ᵐ[Measure.restrict μ t] 0

tactic: intro 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
hg_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn g s
hfg_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, f x ∂μ) = ∫ (x : α) in s, g x ∂μ
t : Set α
ht : MeasurableSet t
hμt : ↑↑μ t ≠ ⊤
⊢ ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, (f - g) x ∂μ) = 0

tactic: rw [integral_sub' (hf_int_finite s hs hμs) (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
hg_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn g s
hfg_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, f x ∂μ) = ∫ (x : α) in s, g x ∂μ
t : Set α
ht : MeasurableSet t
hμt : ↑↑μ t ≠ ⊤
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s < ⊤
⊢ (∫ (x : α) in s, (f - g) x ∂μ) = 0

tactic: exact sub_eq_zero.mpr (hfg_zero 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
hg_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn g s
hfg_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, f x ∂μ) = ∫ (x : α) in s, g x ∂μ
t : Set α
ht : MeasurableSet t
hμt : ↑↑μ t ≠ ⊤
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s < ⊤
⊢ ((∫ (a : α) in s, f a ∂μ) - ∫ (a : α) in s, g a ∂μ) = 0