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
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) ∂Measure.restrict μ t, f x ∈ u
⊢ f =ᵐ[Measure.restrict μ t] 0
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
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 =ᵐ[Measure.restrict μ t] 0
rcases(hf_int_finite t ht hμt.lt_top).aestronglyMeasurable.isSeparable_ae_range with
  ⟨u, u_sep, hu⟩
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
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) ∂Measure.restrict μ t, f x ∈ u
c : Dual ℝ E
⊢ (fun x => ↑c (f x)) =ᵐ[Measure.restrict μ t] 0
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
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) ∂Measure.restrict μ t, f x ∈ u
⊢ f =ᵐ[Measure.restrict μ t] 0
refine' ae_eq_zero_of_forall_dual_of_isSeparable ℝ u_sep (fun c => _) hu
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
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) ∂Measure.restrict μ t, f x ∈ u
c : Dual ℝ E
⊢ ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn (fun x => ↑c (f x)) s

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
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) ∂Measure.restrict μ t, f x ∈ u
c : Dual ℝ E
⊢ ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, ↑c (f x) ∂μ) = 0
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
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) ∂Measure.restrict μ t, f x ∈ u
c : Dual ℝ E
⊢ (fun x => ↑c (f x)) =ᵐ[Measure.restrict μ t] 0
refine' ae_eq_zero_restrict_of_forall_set_integral_eq_zero_real _ _ ht hμt
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
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) ∂Measure.restrict μ t, f x ∈ u
c : Dual ℝ E
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s < ⊤
⊢ IntegrableOn (fun x => ↑c (f x)) s
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
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) ∂Measure.restrict μ t, f x ∈ u
c : Dual ℝ E
⊢ ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn (fun x => ↑c (f x)) s
intro s hs hμs
no goals
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
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) ∂Measure.restrict μ t, f x ∈ u
c : Dual ℝ E
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s < ⊤
⊢ IntegrableOn (fun x => ↑c (f x)) s
exact ContinuousLinearMap.integrable_comp c (hf_int_finite s hs hμs)
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
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) ∂Measure.restrict μ t, f x ∈ u
c : Dual ℝ E
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s < ⊤
⊢ (∫ (x : α) in s, ↑c (f x) ∂μ) = 0
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
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) ∂Measure.restrict μ t, f x ∈ u
c : Dual ℝ E
⊢ ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, ↑c (f x) ∂μ) = 0
intro s hs hμs
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
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) ∂Measure.restrict μ t, f x ∈ u
c : Dual ℝ E
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s < ⊤
⊢ ↑c 0 = 0
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
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) ∂Measure.restrict μ t, f x ∈ u
c : Dual ℝ E
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s < ⊤
⊢ (∫ (x : α) in s, ↑c (f x) ∂μ) = 0
rw [ContinuousLinearMap.integral_comp_comm c (hf_int_finite s hs hμs), hf_zero s hs hμs]
no goals
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
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) ∂Measure.restrict μ t, f x ∈ u
c : Dual ℝ E
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s < ⊤
⊢ ↑c 0 = 0
exact ContinuousLinearMap.map_zero _