Theorem: MeasureTheory.ae_eq_zero_of_forall_setIntegral_isCompact_eq_zero
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✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
p : ℝ≥0∞
β : Type u_3
inst✝⁴ : TopologicalSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
inst✝¹ : SigmaCompactSpace β
inst✝ : R1Space β
μ : Measure β
f : β → E
hf : Integrable f μ
h'f : ∀ (s : Set β), IsCompact s → ∫ (x : β) in s, f x ∂μ = 0
⊢ f =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✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
p : ℝ≥0∞
β : Type u_3
inst✝⁴ : TopologicalSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
inst✝¹ : SigmaCompactSpace β
inst✝ : R1Space β
μ : Measure β
f : β → E
hf : Integrable f μ
h'f : ∀ (s : Set β), IsCompact s → ∫ (x : β) in s, f x ∂μ = 0
s : Set β
hs : IsClosed s
⊢ ∫ (x : β) in s, f x ∂μ = 0
tactic: apply ae_eq_zero_of_forall_setIntegral_isClosed_eq_zero hf (fun s hs ↦ ?_)
state_before: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ✝ : Measure α
s✝ t : Set α
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
p : ℝ≥0∞
β : Type u_3
inst✝⁴ : TopologicalSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
inst✝¹ : SigmaCompactSpace β
inst✝ : R1Space β
μ : Measure β
f : β → E
hf : Integrable f μ
h'f : ∀ (s : Set β), IsCompact s → ∫ (x : β) in s, f x ∂μ = 0
s : Set β
hs : IsClosed s
⊢ ∫ (x : β) in s, f x ∂μ = 0
state_after: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ✝ : Measure α
s✝ t✝ : Set α
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
p : ℝ≥0∞
β : Type u_3
inst✝⁴ : TopologicalSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
inst✝¹ : SigmaCompactSpace β
inst✝ : R1Space β
μ : Measure β
f : β → E
hf : Integrable f μ
h'f : ∀ (s : Set β), IsCompact s → ∫ (x : β) in s, f x ∂μ = 0
s : Set β
hs : IsClosed s
t : ℕ → Set β := fun n => closure (compactCovering β n) ∩ s
⊢ ∫ (x : β) in s, f x ∂μ = 0
tactic: let t : ℕ → Set β := fun n ↦ closure (compactCovering β n) ∩ s
state_before: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ✝ : Measure α
s✝ t✝ : Set α
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
p : ℝ≥0∞
β : Type u_3
inst✝⁴ : TopologicalSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
inst✝¹ : SigmaCompactSpace β
inst✝ : R1Space β
μ : Measure β
f : β → E
hf : Integrable f μ
h'f : ∀ (s : Set β), IsCompact s → ∫ (x : β) in s, f x ∂μ = 0
s : Set β
hs : IsClosed s
t : ℕ → Set β := fun n => closure (compactCovering β n) ∩ s
⊢ ∫ (x : β) in s, f x ∂μ = 0
state_after: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ✝ : Measure α
s✝ t✝ : Set α
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
p : ℝ≥0∞
β : Type u_3
inst✝⁴ : TopologicalSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
inst✝¹ : SigmaCompactSpace β
inst✝ : R1Space β
μ : Measure β
f : β → E
hf : Integrable f μ
h'f : ∀ (s : Set β), IsCompact s → ∫ (x : β) in s, f x ∂μ = 0
s : Set β
hs : IsClosed s
t : ℕ → Set β := fun n => closure (compactCovering β n) ∩ s
⊢ Tendsto (fun n => ∫ (x : β) in t n, f x ∂μ) atTop (𝓝 (∫ (x : β) in s, f x ∂μ))
tactic: suffices H : Tendsto (fun n ↦ ∫ x in t n, f x ∂μ) atTop (𝓝 (∫ x in s, f x ∂μ)) by
    have A : ∀ n, ∫ x in t n, f x ∂μ = 0 :=
      fun n ↦ h'f _ ((isCompact_compactCovering β n).closure.inter_right hs)
    simp_rw [A, tendsto_const_nhds_iff] at H
    exact H.symm
state_before: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ✝ : Measure α
s✝ t✝ : Set α
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
p : ℝ≥0∞
β : Type u_3
inst✝⁴ : TopologicalSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
inst✝¹ : SigmaCompactSpace β
inst✝ : R1Space β
μ : Measure β
f : β → E
hf : Integrable f μ
h'f : ∀ (s : Set β), IsCompact s → ∫ (x : β) in s, f x ∂μ = 0
s : Set β
hs : IsClosed s
t : ℕ → Set β := fun n => closure (compactCovering β n) ∩ s
⊢ Tendsto (fun n => ∫ (x : β) in t n, f x ∂μ) atTop (𝓝 (∫ (x : β) in s, f x ∂μ))
state_after: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ✝ : Measure α
s✝ t✝ : Set α
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
p : ℝ≥0∞
β : Type u_3
inst✝⁴ : TopologicalSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
inst✝¹ : SigmaCompactSpace β
inst✝ : R1Space β
μ : Measure β
f : β → E
hf : Integrable f μ
h'f : ∀ (s : Set β), IsCompact s → ∫ (x : β) in s, f x ∂μ = 0
s : Set β
hs : IsClosed s
t : ℕ → Set β := fun n => closure (compactCovering β n) ∩ s
B : s = ⋃ n, t n
⊢ Tendsto (fun n => ∫ (x : β) in t n, f x ∂μ) atTop (𝓝 (∫ (x : β) in s, f x ∂μ))
tactic: have B : s = ⋃ n, t n := by
    rw [← Set.iUnion_inter, iUnion_closure_compactCovering, Set.univ_inter]
state_before: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ✝ : Measure α
s✝ t✝ : Set α
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
p : ℝ≥0∞
β : Type u_3
inst✝⁴ : TopologicalSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
inst✝¹ : SigmaCompactSpace β
inst✝ : R1Space β
μ : Measure β
f : β → E
hf : Integrable f μ
h'f : ∀ (s : Set β), IsCompact s → ∫ (x : β) in s, f x ∂μ = 0
s : Set β
hs : IsClosed s
t : ℕ → Set β := fun n => closure (compactCovering β n) ∩ s
B : s = ⋃ n, t n
⊢ Tendsto (fun n => ∫ (x : β) in t n, f x ∂μ) atTop (𝓝 (∫ (x : β) in s, f x ∂μ))
state_after: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ✝ : Measure α
s✝ t✝ : Set α
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
p : ℝ≥0∞
β : Type u_3
inst✝⁴ : TopologicalSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
inst✝¹ : SigmaCompactSpace β
inst✝ : R1Space β
μ : Measure β
f : β → E
hf : Integrable f μ
h'f : ∀ (s : Set β), IsCompact s → ∫ (x : β) in s, f x ∂μ = 0
s : Set β
hs : IsClosed s
t : ℕ → Set β := fun n => closure (compactCovering β n) ∩ s
B : s = ⋃ n, t n
⊢ Tendsto (fun n => ∫ (x : β) in t n, f x ∂μ) atTop (𝓝 (∫ (x : β) in ⋃ n, t n, f x ∂μ))
tactic: rw [B]
state_before: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ✝ : Measure α
s✝ t✝ : Set α
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
p : ℝ≥0∞
β : Type u_3
inst✝⁴ : TopologicalSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
inst✝¹ : SigmaCompactSpace β
inst✝ : R1Space β
μ : Measure β
f : β → E
hf : Integrable f μ
h'f : ∀ (s : Set β), IsCompact s → ∫ (x : β) in s, f x ∂μ = 0
s : Set β
hs : IsClosed s
t : ℕ → Set β := fun n => closure (compactCovering β n) ∩ s
B : s = ⋃ n, t n
⊢ Tendsto (fun n => ∫ (x : β) in t n, f x ∂μ) atTop (𝓝 (∫ (x : β) in ⋃ n, t n, f x ∂μ))
state_after: case hsm
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ✝ : Measure α
s✝ t✝ : Set α
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
p : ℝ≥0∞
β : Type u_3
inst✝⁴ : TopologicalSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
inst✝¹ : SigmaCompactSpace β
inst✝ : R1Space β
μ : Measure β
f : β → E
hf : Integrable f μ
h'f : ∀ (s : Set β), IsCompact s → ∫ (x : β) in s, f x ∂μ = 0
s : Set β
hs : IsClosed s
t : ℕ → Set β := fun n => closure (compactCovering β n) ∩ s
B : s = ⋃ n, t n
⊢ ∀ (i : ℕ), MeasurableSet (t i)

case h_mono
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ✝ : Measure α
s✝ t✝ : Set α
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
p : ℝ≥0∞
β : Type u_3
inst✝⁴ : TopologicalSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
inst✝¹ : SigmaCompactSpace β
inst✝ : R1Space β
μ : Measure β
f : β → E
hf : Integrable f μ
h'f : ∀ (s : Set β), IsCompact s → ∫ (x : β) in s, f x ∂μ = 0
s : Set β
hs : IsClosed s
t : ℕ → Set β := fun n => closure (compactCovering β n) ∩ s
B : s = ⋃ n, t n
⊢ Monotone t

case hfi
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ✝ : Measure α
s✝ t✝ : Set α
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
p : ℝ≥0∞
β : Type u_3
inst✝⁴ : TopologicalSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
inst✝¹ : SigmaCompactSpace β
inst✝ : R1Space β
μ : Measure β
f : β → E
hf : Integrable f μ
h'f : ∀ (s : Set β), IsCompact s → ∫ (x : β) in s, f x ∂μ = 0
s : Set β
hs : IsClosed s
t : ℕ → Set β := fun n => closure (compactCovering β n) ∩ s
B : s = ⋃ n, t n
⊢ IntegrableOn f (⋃ n, t n) μ
tactic: apply tendsto_setIntegral_of_monotone