Theorem: MeasureTheory.AEFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero
File: Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean

tactic: let t := hf.sigmaFiniteSet
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
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, f x ∂μ) = 0
hf : AEFinStronglyMeasurable f μ
⊢ f =ᵐ[μ] 0

tactic: suffices : f =ᵐ[μ.restrict t] 0
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
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, f x ∂μ) = 0
hf : AEFinStronglyMeasurable f μ
t : Set α := sigmaFiniteSet hf
⊢ f =ᵐ[μ] 0

tactic: exact ae_of_ae_restrict_of_ae_restrict_compl _ this hf.ae_eq_zero_compl
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
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, f x ∂μ) = 0
hf : AEFinStronglyMeasurable f μ
t : Set α := sigmaFiniteSet hf
this : f =ᵐ[Measure.restrict μ t] 0
⊢ f =ᵐ[μ] 0

case this
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
hf : AEFinStronglyMeasurable f μ
t : Set α := sigmaFiniteSet hf
⊢ f =ᵐ[Measure.restrict μ t] 0

tactic: haveI : SigmaFinite (μ.restrict t) := hf.sigmaFinite_restrict
state_before: case this
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
hf : AEFinStronglyMeasurable f μ
t : Set α := sigmaFiniteSet hf
⊢ f =ᵐ[Measure.restrict μ t] 0

tactic: refine' ae_eq_zero_of_forall_set_integral_eq_of_sigmaFinite _ _
state_before: case this
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
hf : AEFinStronglyMeasurable f μ
t : Set α := sigmaFiniteSet hf
this : SigmaFinite (Measure.restrict μ t)
⊢ f =ᵐ[Measure.restrict μ t] 0

tactic: intro s hs hμs
state_before: case this.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
hf : AEFinStronglyMeasurable f μ
t : Set α := sigmaFiniteSet hf
this : SigmaFinite (Measure.restrict μ t)
⊢ ∀ (s : Set α), MeasurableSet s → ↑↑(Measure.restrict μ t) s < ⊤ → IntegrableOn f s

tactic: rw [IntegrableOn, Measure.restrict_restrict hs]
state_before: case this.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
hf : AEFinStronglyMeasurable f μ
t : Set α := sigmaFiniteSet hf
this : SigmaFinite (Measure.restrict μ t)
s : Set α
hs : MeasurableSet s
hμs : ↑↑(Measure.restrict μ t) s < ⊤
⊢ IntegrableOn f s

tactic: rw [Measure.restrict_apply hs] at hμs
state_before: case this.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
hf : AEFinStronglyMeasurable f μ
t : Set α := sigmaFiniteSet hf
this : SigmaFinite (Measure.restrict μ t)
s : Set α
hs : MeasurableSet s
hμs : ↑↑(Measure.restrict μ t) s < ⊤
⊢ Integrable f

tactic: exact hf_int_finite _ (hs.inter hf.measurableSet) hμs
state_before: case this.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
hf : AEFinStronglyMeasurable f μ
t : Set α := sigmaFiniteSet hf
this : SigmaFinite (Measure.restrict μ t)
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ (s ∩ t) < ⊤
⊢ Integrable f

tactic: intro s hs hμs
state_before: case this.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
hf : AEFinStronglyMeasurable f μ
t : Set α := sigmaFiniteSet hf
this : SigmaFinite (Measure.restrict μ t)
⊢ ∀ (s : Set α), MeasurableSet s → ↑↑(Measure.restrict μ t) s < ⊤ → (∫ (x : α) in s, f x ∂Measure.restrict μ t) = 0

tactic: rw [Measure.restrict_restrict hs]
state_before: case this.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
hf : AEFinStronglyMeasurable f μ
t : Set α := sigmaFiniteSet hf
this : SigmaFinite (Measure.restrict μ t)
s : Set α
hs : MeasurableSet s
hμs : ↑↑(Measure.restrict μ t) s < ⊤
⊢ (∫ (x : α) in s, f x ∂Measure.restrict μ t) = 0

tactic: rw [Measure.restrict_apply hs] at hμs
state_before: case this.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
hf : AEFinStronglyMeasurable f μ
t : Set α := sigmaFiniteSet hf
this : SigmaFinite (Measure.restrict μ t)
s : Set α
hs : MeasurableSet s
hμs : ↑↑(Measure.restrict μ t) s < ⊤
⊢ (∫ (x : α) in s ∩ t, f x ∂μ) = 0

tactic: exact hf_zero _ (hs.inter hf.measurableSet) hμs
state_before: case this.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
hf : AEFinStronglyMeasurable f μ
t : Set α := sigmaFiniteSet hf
this : SigmaFinite (Measure.restrict μ t)
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ (s ∩ t) < ⊤
⊢ (∫ (x : α) in s ∩ t, f x ∂μ) = 0