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

tactic: obtain ⟨t, ht_meas, htf_zero, htμ⟩ := hf.exists_set_sigmaFinite
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∞
hm : m ≤ m0
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 : FinStronglyMeasurable f (Measure.trim μ hm)
⊢ f =ᵐ[μ] 0

tactic: haveI : SigmaFinite ((μ.restrict t).trim hm) := by rwa [restrict_trim hm μ ht_meas] at htμ
state_before: case intro.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∞
hm : m ≤ m0
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 : FinStronglyMeasurable f (Measure.trim μ hm)
t : Set α
ht_meas : MeasurableSet t
htf_zero : ∀ (x : α), x ∈ tᶜ → f x = 0
htμ : SigmaFinite (Measure.restrict (Measure.trim μ hm) t)
⊢ f =ᵐ[μ] 0

tactic: have htf_zero : f =ᵐ[μ.restrict (tᶜ)] 0 := by
  rw [EventuallyEq, ae_restrict_iff' (MeasurableSet.compl (hm _ ht_meas))]
  exact eventually_of_forall htf_zero
state_before: case intro.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∞
hm : m ≤ m0
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 : FinStronglyMeasurable f (Measure.trim μ hm)
t : Set α
ht_meas : MeasurableSet t
htf_zero : ∀ (x : α), x ∈ tᶜ → f x = 0
htμ : SigmaFinite (Measure.restrict (Measure.trim μ hm) t)
this : SigmaFinite (Measure.trim (Measure.restrict μ t) hm)
⊢ f =ᵐ[μ] 0

tactic: have hf_meas_m : StronglyMeasurable[m] f := hf.stronglyMeasurable
state_before: case intro.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∞
hm : m ≤ m0
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 : FinStronglyMeasurable f (Measure.trim μ hm)
t : Set α
ht_meas : MeasurableSet t
htf_zero✝ : ∀ (x : α), x ∈ tᶜ → f x = 0
htμ : SigmaFinite (Measure.restrict (Measure.trim μ hm) t)
this : SigmaFinite (Measure.trim (Measure.restrict μ t) hm)
htf_zero : f =ᵐ[Measure.restrict μ (tᶜ)] 0
⊢ f =ᵐ[μ] 0

tactic: suffices : f =ᵐ[μ.restrict t] 0
state_before: case intro.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∞
hm : m ≤ m0
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 : FinStronglyMeasurable f (Measure.trim μ hm)
t : Set α
ht_meas : MeasurableSet t
htf_zero✝ : ∀ (x : α), x ∈ tᶜ → f x = 0
htμ : SigmaFinite (Measure.restrict (Measure.trim μ hm) t)
this : SigmaFinite (Measure.trim (Measure.restrict μ t) hm)
htf_zero : f =ᵐ[Measure.restrict μ (tᶜ)] 0
hf_meas_m : StronglyMeasurable f
⊢ f =ᵐ[μ] 0

tactic: exact ae_of_ae_restrict_of_ae_restrict_compl _ this htf_zero
state_before: case intro.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∞
hm : m ≤ m0
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 : FinStronglyMeasurable f (Measure.trim μ hm)
t : Set α
ht_meas : MeasurableSet t
htf_zero✝ : ∀ (x : α), x ∈ tᶜ → f x = 0
htμ : SigmaFinite (Measure.restrict (Measure.trim μ hm) t)
this✝ : SigmaFinite (Measure.trim (Measure.restrict μ t) hm)
htf_zero : f =ᵐ[Measure.restrict μ (tᶜ)] 0
hf_meas_m : StronglyMeasurable f
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∞
hm : m ≤ m0
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 : FinStronglyMeasurable f (Measure.trim μ hm)
t : Set α
ht_meas : MeasurableSet t
htf_zero✝ : ∀ (x : α), x ∈ tᶜ → f x = 0
htμ : SigmaFinite (Measure.restrict (Measure.trim μ hm) t)
this : SigmaFinite (Measure.trim (Measure.restrict μ t) hm)
htf_zero : f =ᵐ[Measure.restrict μ (tᶜ)] 0
hf_meas_m : StronglyMeasurable f
⊢ f =ᵐ[Measure.restrict μ t] 0

tactic: refine' measure_eq_zero_of_trim_eq_zero hm _
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∞
hm : m ≤ m0
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 : FinStronglyMeasurable f (Measure.trim μ hm)
t : Set α
ht_meas : MeasurableSet t
htf_zero✝ : ∀ (x : α), x ∈ tᶜ → f x = 0
htμ : SigmaFinite (Measure.restrict (Measure.trim μ hm) t)
this : SigmaFinite (Measure.trim (Measure.restrict μ t) hm)
htf_zero : f =ᵐ[Measure.restrict μ (tᶜ)] 0
hf_meas_m : StronglyMeasurable f
⊢ 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∞
hm : m ≤ m0
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 : FinStronglyMeasurable f (Measure.trim μ hm)
t : Set α
ht_meas : MeasurableSet t
htf_zero✝ : ∀ (x : α), x ∈ tᶜ → f x = 0
htμ : SigmaFinite (Measure.restrict (Measure.trim μ hm) t)
this : SigmaFinite (Measure.trim (Measure.restrict μ t) hm)
htf_zero : f =ᵐ[Measure.restrict μ (tᶜ)] 0
hf_meas_m : StronglyMeasurable f
⊢ ↑↑(Measure.trim (Measure.restrict μ t) hm) ({x | (fun x => f x = OfNat.ofNat 0 x) x}ᶜ) = 0

tactic: rwa [restrict_trim hm μ ht_meas] at htμ
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∞
hm : m ≤ m0
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 : FinStronglyMeasurable f (Measure.trim μ hm)
t : Set α
ht_meas : MeasurableSet t
htf_zero : ∀ (x : α), x ∈ tᶜ → f x = 0
htμ : SigmaFinite (Measure.restrict (Measure.trim μ hm) t)
⊢ SigmaFinite (Measure.trim (Measure.restrict μ t) hm)

tactic: rw [EventuallyEq, ae_restrict_iff' (MeasurableSet.compl (hm _ ht_meas))]
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∞
hm : m ≤ m0
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 : FinStronglyMeasurable f (Measure.trim μ hm)
t : Set α
ht_meas : MeasurableSet t
htf_zero : ∀ (x : α), x ∈ tᶜ → f x = 0
htμ : SigmaFinite (Measure.restrict (Measure.trim μ hm) t)
this : SigmaFinite (Measure.trim (Measure.restrict μ t) hm)
⊢ f =ᵐ[Measure.restrict μ (tᶜ)] 0

tactic: exact eventually_of_forall htf_zero
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∞
hm : m ≤ m0
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 : FinStronglyMeasurable f (Measure.trim μ hm)
t : Set α
ht_meas : MeasurableSet t
htf_zero : ∀ (x : α), x ∈ tᶜ → f x = 0
htμ : SigmaFinite (Measure.restrict (Measure.trim μ hm) t)
this : SigmaFinite (Measure.trim (Measure.restrict μ t) hm)
⊢ ∀ᵐ (x : α) ∂μ, x ∈ tᶜ → f x = OfNat.ofNat 0 x

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∞
hm : m ≤ m0
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 : FinStronglyMeasurable f (Measure.trim μ hm)
t : Set α
ht_meas : MeasurableSet t
htf_zero✝ : ∀ (x : α), x ∈ tᶜ → f x = 0
htμ : SigmaFinite (Measure.restrict (Measure.trim μ hm) t)
this : SigmaFinite (Measure.trim (Measure.restrict μ t) hm)
htf_zero : f =ᵐ[Measure.restrict μ (tᶜ)] 0
hf_meas_m : StronglyMeasurable f
⊢ ∀ (s : Set α), MeasurableSet s → ↑↑(Measure.trim (Measure.restrict μ t) hm) s < ⊤ → IntegrableOn (fun x => f x) s

tactic: unfold IntegrableOn
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∞
hm : m ≤ m0
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 : FinStronglyMeasurable f (Measure.trim μ hm)
t : Set α
ht_meas : MeasurableSet t
htf_zero✝ : ∀ (x : α), x ∈ tᶜ → f x = 0
htμ : SigmaFinite (Measure.restrict (Measure.trim μ hm) t)
this : SigmaFinite (Measure.trim (Measure.restrict μ t) hm)
htf_zero : f =ᵐ[Measure.restrict μ (tᶜ)] 0
hf_meas_m : StronglyMeasurable f
s : Set α
hs : MeasurableSet s
hμs : ↑↑(Measure.trim (Measure.restrict μ t) hm) s < ⊤
⊢ IntegrableOn (fun x => f x) s

tactic: rw [restrict_trim hm (μ.restrict t) hs, Measure.restrict_restrict (hm s 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∞
hm : m ≤ m0
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 : FinStronglyMeasurable f (Measure.trim μ hm)
t : Set α
ht_meas : MeasurableSet t
htf_zero✝ : ∀ (x : α), x ∈ tᶜ → f x = 0
htμ : SigmaFinite (Measure.restrict (Measure.trim μ hm) t)
this : SigmaFinite (Measure.trim (Measure.restrict μ t) hm)
htf_zero : f =ᵐ[Measure.restrict μ (tᶜ)] 0
hf_meas_m : StronglyMeasurable f
s : Set α
hs : MeasurableSet s
hμs : ↑↑(Measure.trim (Measure.restrict μ t) hm) s < ⊤
⊢ Integrable fun x => f x

tactic: rw [← restrict_trim hm μ ht_meas, Measure.restrict_apply hs,
  trim_measurableSet_eq hm (hs.inter ht_meas)] 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∞
hm : m ≤ m0
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 : FinStronglyMeasurable f (Measure.trim μ hm)
t : Set α
ht_meas : MeasurableSet t
htf_zero✝ : ∀ (x : α), x ∈ tᶜ → f x = 0
htμ : SigmaFinite (Measure.restrict (Measure.trim μ hm) t)
this : SigmaFinite (Measure.trim (Measure.restrict μ t) hm)
htf_zero : f =ᵐ[Measure.restrict μ (tᶜ)] 0
hf_meas_m : StronglyMeasurable f
s : Set α
hs : MeasurableSet s
hμs : ↑↑(Measure.trim (Measure.restrict μ t) hm) s < ⊤
⊢ Integrable fun x => f x

tactic: refine' Integrable.trim hm _ hf_meas_m
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∞
hm : m ≤ m0
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 : FinStronglyMeasurable f (Measure.trim μ hm)
t : Set α
ht_meas : MeasurableSet t
htf_zero✝ : ∀ (x : α), x ∈ tᶜ → f x = 0
htμ : SigmaFinite (Measure.restrict (Measure.trim μ hm) t)
this : SigmaFinite (Measure.trim (Measure.restrict μ t) hm)
htf_zero : f =ᵐ[Measure.restrict μ (tᶜ)] 0
hf_meas_m : StronglyMeasurable f
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ (s ∩ t) < ⊤
⊢ Integrable fun x => f x

tactic: exact hf_int_finite _ (hs.inter ht_meas) 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∞
hm : m ≤ m0
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 : FinStronglyMeasurable f (Measure.trim μ hm)
t : Set α
ht_meas : MeasurableSet t
htf_zero✝ : ∀ (x : α), x ∈ tᶜ → f x = 0
htμ : SigmaFinite (Measure.restrict (Measure.trim μ hm) t)
this : SigmaFinite (Measure.trim (Measure.restrict μ t) hm)
htf_zero : f =ᵐ[Measure.restrict μ (tᶜ)] 0
hf_meas_m : StronglyMeasurable f
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ (s ∩ t) < ⊤
⊢ Integrable fun x => f x

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∞
hm : m ≤ m0
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 : FinStronglyMeasurable f (Measure.trim μ hm)
t : Set α
ht_meas : MeasurableSet t
htf_zero✝ : ∀ (x : α), x ∈ tᶜ → f x = 0
htμ : SigmaFinite (Measure.restrict (Measure.trim μ hm) t)
this : SigmaFinite (Measure.trim (Measure.restrict μ t) hm)
htf_zero : f =ᵐ[Measure.restrict μ (tᶜ)] 0
hf_meas_m : StronglyMeasurable f
⊢ ∀ (s : Set α),
    MeasurableSet s →
      ↑↑(Measure.trim (Measure.restrict μ t) hm) s < ⊤ →
        (∫ (x : α) in s, f x ∂Measure.trim (Measure.restrict μ t) hm) = 0

tactic: rw [restrict_trim hm (μ.restrict t) hs, Measure.restrict_restrict (hm s 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∞
hm : m ≤ m0
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 : FinStronglyMeasurable f (Measure.trim μ hm)
t : Set α
ht_meas : MeasurableSet t
htf_zero✝ : ∀ (x : α), x ∈ tᶜ → f x = 0
htμ : SigmaFinite (Measure.restrict (Measure.trim μ hm) t)
this : SigmaFinite (Measure.trim (Measure.restrict μ t) hm)
htf_zero : f =ᵐ[Measure.restrict μ (tᶜ)] 0
hf_meas_m : StronglyMeasurable f
s : Set α
hs : MeasurableSet s
hμs : ↑↑(Measure.trim (Measure.restrict μ t) hm) s < ⊤
⊢ (∫ (x : α) in s, f x ∂Measure.trim (Measure.restrict μ t) hm) = 0

tactic: rw [← restrict_trim hm μ ht_meas, Measure.restrict_apply hs,
  trim_measurableSet_eq hm (hs.inter ht_meas)] 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∞
hm : m ≤ m0
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 : FinStronglyMeasurable f (Measure.trim μ hm)
t : Set α
ht_meas : MeasurableSet t
htf_zero✝ : ∀ (x : α), x ∈ tᶜ → f x = 0
htμ : SigmaFinite (Measure.restrict (Measure.trim μ hm) t)
this : SigmaFinite (Measure.trim (Measure.restrict μ t) hm)
htf_zero : f =ᵐ[Measure.restrict μ (tᶜ)] 0
hf_meas_m : StronglyMeasurable f
s : Set α
hs : MeasurableSet s
hμs : ↑↑(Measure.trim (Measure.restrict μ t) hm) s < ⊤
⊢ (∫ (x : α), f x ∂Measure.trim (Measure.restrict μ (s ∩ t)) hm) = 0

tactic: rw [← integral_trim hm hf_meas_m]
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∞
hm : m ≤ m0
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 : FinStronglyMeasurable f (Measure.trim μ hm)
t : Set α
ht_meas : MeasurableSet t
htf_zero✝ : ∀ (x : α), x ∈ tᶜ → f x = 0
htμ : SigmaFinite (Measure.restrict (Measure.trim μ hm) t)
this : SigmaFinite (Measure.trim (Measure.restrict μ t) hm)
htf_zero : f =ᵐ[Measure.restrict μ (tᶜ)] 0
hf_meas_m : StronglyMeasurable f
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ (s ∩ t) < ⊤
⊢ (∫ (x : α), f x ∂Measure.trim (Measure.restrict μ (s ∩ t)) hm) = 0

tactic: exact hf_zero _ (hs.inter ht_meas) 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∞
hm : m ≤ m0
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 : FinStronglyMeasurable f (Measure.trim μ hm)
t : Set α
ht_meas : MeasurableSet t
htf_zero✝ : ∀ (x : α), x ∈ tᶜ → f x = 0
htμ : SigmaFinite (Measure.restrict (Measure.trim μ hm) t)
this : SigmaFinite (Measure.trim (Measure.restrict μ t) hm)
htf_zero : f =ᵐ[Measure.restrict μ (tᶜ)] 0
hf_meas_m : StronglyMeasurable f
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ (s ∩ t) < ⊤
⊢ (∫ (x : α) in s ∩ t, f x ∂μ) = 0