Theorem: MeasureTheory.ae_eq_zero_restrict_of_forall_setIntegral_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✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ≥0∞
f : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ = 0
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ⊤
⊢ f =ᶠ[ae (μ.restrict t)] 0
state_after: case intro.intro
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 : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ = 0
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ⊤
u : Set E
u_sep : IsSeparable u
hu : ∀ᵐ (x : α) ∂μ.restrict t, f x ∈ u
⊢ f =ᶠ[ae (μ.restrict t)] 0
tactic: rcases (hf_int_finite t ht hμt.lt_top).aestronglyMeasurable.isSeparable_ae_range with
    ⟨u, u_sep, hu⟩
state_before: case intro.intro
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 : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ = 0
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ⊤
u : Set E
u_sep : IsSeparable u
hu : ∀ᵐ (x : α) ∂μ.restrict t, f x ∈ u
⊢ f =ᶠ[ae (μ.restrict t)] 0
state_after: case intro.intro
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 : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ = 0
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ⊤
u : Set E
u_sep : IsSeparable u
hu : ∀ᵐ (x : α) ∂μ.restrict t, f x ∈ u
c : Dual ℝ E
⊢ (fun x => c (f x)) =ᶠ[ae (μ.restrict t)] 0
tactic: refine ae_eq_zero_of_forall_dual_of_isSeparable ℝ u_sep (fun c => ?_) hu
state_before: case intro.intro
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 : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ = 0
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ⊤
u : Set E
u_sep : IsSeparable u
hu : ∀ᵐ (x : α) ∂μ.restrict t, f x ∈ u
c : Dual ℝ E
⊢ (fun x => c (f x)) =ᶠ[ae (μ.restrict t)] 0
state_after: case intro.intro.refine_1
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 : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ = 0
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ⊤
u : Set E
u_sep : IsSeparable u
hu : ∀ᵐ (x : α) ∂μ.restrict t, f x ∈ u
c : Dual ℝ E
⊢ ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn (fun x => c (f x)) s μ

case intro.intro.refine_2
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 : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ = 0
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ⊤
u : Set E
u_sep : IsSeparable u
hu : ∀ᵐ (x : α) ∂μ.restrict t, f x ∈ u
c : Dual ℝ E
⊢ ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, c (f x) ∂μ = 0
tactic: refine ae_eq_zero_restrict_of_forall_setIntegral_eq_zero_real ?_ ?_ ht hμt