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

tactic: let S := spanningSets μ
state_before: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f : α → E
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
⊢ f =ᵐ[μ] 0

tactic: rw [← @Measure.restrict_univ _ _ μ, ← iUnion_spanningSets μ, EventuallyEq, ae_iff,
  Measure.restrict_apply' (MeasurableSet.iUnion (measurable_spanningSets μ))]
state_before: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f : α → E
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
S : ℕ → Set α := spanningSets μ
⊢ f =ᵐ[μ] 0

tactic: rw [Set.inter_iUnion, measure_iUnion_null_iff]
state_before: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f : α → E
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
S : ℕ → Set α := spanningSets μ
⊢ ↑↑μ ({a | ¬f a = OfNat.ofNat 0 a} ∩ ⋃ (b : ℕ), spanningSets μ b) = 0

tactic: intro n
state_before: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f : α → E
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
S : ℕ → Set α := spanningSets μ
⊢ ∀ (i : ℕ), ↑↑μ ({a | ¬f a = OfNat.ofNat 0 a} ∩ spanningSets μ i) = 0

tactic: have h_meas_n : MeasurableSet (S n) := measurable_spanningSets μ n
state_before: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f : α → E
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
S : ℕ → Set α := spanningSets μ
n : ℕ
⊢ ↑↑μ ({a | ¬f a = OfNat.ofNat 0 a} ∩ spanningSets μ n) = 0

tactic: have hμn : μ (S n) < ∞ := measure_spanningSets_lt_top μ n
state_before: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f : α → E
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
S : ℕ → Set α := spanningSets μ
n : ℕ
h_meas_n : MeasurableSet (S n)
⊢ ↑↑μ ({a | ¬f a = OfNat.ofNat 0 a} ∩ spanningSets μ n) = 0

tactic: rw [← Measure.restrict_apply' h_meas_n]
state_before: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f : α → E
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
S : ℕ → Set α := spanningSets μ
n : ℕ
h_meas_n : MeasurableSet (S n)
hμn : ↑↑μ (S n) < ⊤
⊢ ↑↑μ ({a | ¬f a = OfNat.ofNat 0 a} ∩ spanningSets μ n) = 0

tactic: exact ae_eq_zero_restrict_of_forall_set_integral_eq_zero hf_int_finite hf_zero h_meas_n hμn.ne
state_before: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f : α → E
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
S : ℕ → Set α := spanningSets μ
n : ℕ
h_meas_n : MeasurableSet (S n)
hμn : ↑↑μ (S n) < ⊤
⊢ ↑↑(Measure.restrict μ (S n)) {a | ¬f a = OfNat.ofNat 0 a} = 0