Theorem: MeasureTheory.integral_union_eq_left_of_ae_aux
File: Mathlib/MeasureTheory/Integral/SetIntegral.lean

Proof steps:
state_before: X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s t : Set X
μ ν : Measure X
l l' : Filter X
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
haux : StronglyMeasurable f
H : IntegrableOn f (s ∪ t) μ
⊢ ∫ (x : X) in s ∪ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ
state_after: X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s t : Set X
μ ν : Measure X
l l' : Filter X
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
haux : StronglyMeasurable f
H : IntegrableOn f (s ∪ t) μ
k : Set X := f ⁻¹' {0}
⊢ ∫ (x : X) in s ∪ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ
tactic: let k := f ⁻¹' {0}
state_before: X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s t : Set X
μ ν : Measure X
l l' : Filter X
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
haux : StronglyMeasurable f
H : IntegrableOn f (s ∪ t) μ
k : Set X := f ⁻¹' {0}
⊢ ∫ (x : X) in s ∪ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ
state_after: X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s t : Set X
μ ν : Measure X
l l' : Filter X
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
haux : StronglyMeasurable f
H : IntegrableOn f (s ∪ t) μ
k : Set X := f ⁻¹' {0}
hk : MeasurableSet k
⊢ ∫ (x : X) in s ∪ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ
tactic: have hk : MeasurableSet k := by borelize E; exact haux.measurable (measurableSet_singleton _)
state_before: X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s t : Set X
μ ν : Measure X
l l' : Filter X
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
haux : StronglyMeasurable f
H : IntegrableOn f (s ∪ t) μ
k : Set X := f ⁻¹' {0}
hk : MeasurableSet k
⊢ ∫ (x : X) in s ∪ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ
state_after: X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s t : Set X
μ ν : Measure X
l l' : Filter X
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
haux : StronglyMeasurable f
H : IntegrableOn f (s ∪ t) μ
k : Set X := f ⁻¹' {0}
hk : MeasurableSet k
h's : IntegrableOn f s μ
⊢ ∫ (x : X) in s ∪ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ
tactic: have h's : IntegrableOn f s μ := H.mono subset_union_left le_rfl
state_before: X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s t : Set X
μ ν : Measure X
l l' : Filter X
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
haux : StronglyMeasurable f
H : IntegrableOn f (s ∪ t) μ
k : Set X := f ⁻¹' {0}
hk : MeasurableSet k
h's : IntegrableOn f s μ
⊢ ∫ (x : X) in s ∪ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ
state_after: X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s t : Set X
μ ν : Measure X
l l' : Filter X
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
haux : StronglyMeasurable f
H : IntegrableOn f (s ∪ t) μ
k : Set X := f ⁻¹' {0}
hk : MeasurableSet k
h's : IntegrableOn f s μ
A : ∀ (u : Set X), ∫ (x : X) in u ∩ k, f x ∂μ = 0
⊢ ∫ (x : X) in s ∪ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ
tactic: have A : ∀ u : Set X, ∫ x in u ∩ k, f x ∂μ = 0 := fun u =>
    setIntegral_eq_zero_of_forall_eq_zero fun x hx => hx.2
state_before: X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s t : Set X
μ ν : Measure X
l l' : Filter X
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
haux : StronglyMeasurable f
H : IntegrableOn f (s ∪ t) μ
k : Set X := f ⁻¹' {0}
hk : MeasurableSet k
h's : IntegrableOn f s μ
A : ∀ (u : Set X), ∫ (x : X) in u ∩ k, f x ∂μ = 0
⊢ ∫ (x : X) in s ∪ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ
state_after: X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s t : Set X
μ ν : Measure X
l l' : Filter X
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
haux : StronglyMeasurable f
H : IntegrableOn f (s ∪ t) μ
k : Set X := f ⁻¹' {0}
hk : MeasurableSet k
h's : IntegrableOn f s μ
A : ∀ (u : Set X), ∫ (x : X) in u ∩ k, f x ∂μ = 0
⊢ ∫ (x : X) in t \ k ∪ s \ k, f x ∂μ = ∫ (x : X) in s \ k, f x ∂μ
tactic: rw [← integral_inter_add_diff hk h's, ← integral_inter_add_diff hk H, A, A, zero_add, zero_add,
    union_diff_distrib, union_comm]
state_before: X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s t : Set X
μ ν : Measure X
l l' : Filter X
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
haux : StronglyMeasurable f
H : IntegrableOn f (s ∪ t) μ
k : Set X := f ⁻¹' {0}
hk : MeasurableSet k
h's : IntegrableOn f s μ
A : ∀ (u : Set X), ∫ (x : X) in u ∩ k, f x ∂μ = 0
⊢ ∫ (x : X) in t \ k ∪ s \ k, f x ∂μ = ∫ (x : X) in s \ k, f x ∂μ
state_after: case hst
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s t : Set X
μ ν : Measure X
l l' : Filter X
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
haux : StronglyMeasurable f
H : IntegrableOn f (s ∪ t) μ
k : Set X := f ⁻¹' {0}
hk : MeasurableSet k
h's : IntegrableOn f s μ
A : ∀ (u : Set X), ∫ (x : X) in u ∩ k, f x ∂μ = 0
⊢ t \ k ∪ s \ k =ᶠ[ae μ] s \ k
tactic: apply setIntegral_congr_set_ae
state_before: case hst
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s t : Set X
μ ν : Measure X
l l' : Filter X
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
haux : StronglyMeasurable f
H : IntegrableOn f (s ∪ t) μ
k : Set X := f ⁻¹' {0}
hk : MeasurableSet k
h's : IntegrableOn f s μ
A : ∀ (u : Set X), ∫ (x : X) in u ∩ k, f x ∂μ = 0
⊢ t \ k ∪ s \ k =ᶠ[ae μ] s \ k
state_after: case hst
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s t : Set X
μ ν : Measure X
l l' : Filter X
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
haux : StronglyMeasurable f
H : IntegrableOn f (s ∪ t) μ
k : Set X := f ⁻¹' {0}
hk : MeasurableSet k
h's : IntegrableOn f s μ
A : ∀ (u : Set X), ∫ (x : X) in u ∩ k, f x ∂μ = 0
⊢ μ ((t \ k) \ (s \ k)) = 0
tactic: rw [union_ae_eq_right]
state_before: case hst
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s t : Set X
μ ν : Measure X
l l' : Filter X
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
haux : StronglyMeasurable f
H : IntegrableOn f (s ∪ t) μ
k : Set X := f ⁻¹' {0}
hk : MeasurableSet k
h's : IntegrableOn f s μ
A : ∀ (u : Set X), ∫ (x : X) in u ∩ k, f x ∂μ = 0
⊢ μ ((t \ k) \ (s \ k)) = 0
state_after: case hst
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s t : Set X
μ ν : Measure X
l l' : Filter X
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
haux : StronglyMeasurable f
H : IntegrableOn f (s ∪ t) μ
k : Set X := f ⁻¹' {0}
hk : MeasurableSet k
h's : IntegrableOn f s μ
A : ∀ (u : Set X), ∫ (x : X) in u ∩ k, f x ∂μ = 0
⊢ μ (t \ k) = 0
tactic: apply measure_mono_null diff_subset
state_before: case hst
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s t : Set X
μ ν : Measure X
l l' : Filter X
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
haux : StronglyMeasurable f
H : IntegrableOn f (s ∪ t) μ
k : Set X := f ⁻¹' {0}
hk : MeasurableSet k
h's : IntegrableOn f s μ
A : ∀ (u : Set X), ∫ (x : X) in u ∩ k, f x ∂μ = 0
⊢ μ (t \ k) = 0
state_after: case hst
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s t : Set X
μ ν : Measure X
l l' : Filter X
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
haux : StronglyMeasurable f
H : IntegrableOn f (s ∪ t) μ
k : Set X := f ⁻¹' {0}
hk : MeasurableSet k
h's : IntegrableOn f s μ
A : ∀ (u : Set X), ∫ (x : X) in u ∩ k, f x ∂μ = 0
⊢ ∀ᵐ (a : X) ∂μ, a ∉ t \ k
tactic: rw [measure_zero_iff_ae_nmem]
state_before: case hst
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s t : Set X
μ ν : Measure X
l l' : Filter X
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
haux : StronglyMeasurable f
H : IntegrableOn f (s ∪ t) μ
k : Set X := f ⁻¹' {0}
hk : MeasurableSet k
h's : IntegrableOn f s μ
A : ∀ (u : Set X), ∫ (x : X) in u ∩ k, f x ∂μ = 0
⊢ ∀ᵐ (a : X) ∂μ, a ∉ t \ k
state_after: no goals
tactic: filter_upwards [ae_imp_of_ae_restrict ht_eq] with x hx h'x using h'x.2 (hx h'x.1)