mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
β : Type u_7
mβ : MeasurableSpace β
inst✝ : SFinite μ
hT : DominatedFinMeasAdditive μ T C
h'T : ∀ (s : Set (β × α)), MeasurableSet s → StronglyMeasurable fun x ↦ T (Prod.mk x ⁻¹' s)
f : β → α → E
hf : StronglyMeasurable (Function.uncurry f)
hF : CompleteSpace F
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : SeparableSpace ↑(range (Function.uncurry f) ∪ {0})
s : ℕ → SimpleFunc (β × α) E := SimpleFunc.approxOn (Function.uncurry f) ⋯ (range (Function.uncurry f) ∪ {0}) 0 ⋯
s' : ℕ → β → SimpleFunc α E := fun n x ↦ (s n).comp (Prod.mk x) ⋯
f' : ℕ → β → F := fun n ↦ {x | Integrable (f x) μ}.indicator fun x ↦ SimpleFunc.setToSimpleFunc T (s' n x)
n : ℕ
x : β
⊢ {x ∈ (s' n x).range | x ≠ 0} ⊆ (s n).range

Tactic:
;

Goal state after:
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
β : Type u_7
mβ : MeasurableSpace β
inst✝ : SFinite μ
hT : DominatedFinMeasAdditive μ T C
h'T : ∀ (s : Set (β × α)), MeasurableSet s → StronglyMeasurable fun x ↦ T (Prod.mk x ⁻¹' s)
f : β → α → E
hf : StronglyMeasurable (Function.uncurry f)
hF : CompleteSpace F
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : SeparableSpace ↑(range (Function.uncurry f) ∪ {0})
s : ℕ → SimpleFunc (β × α) E := SimpleFunc.approxOn (Function.uncurry f) ⋯ (range (Function.uncurry f) ∪ {0}) 0 ⋯
s' : ℕ → β → SimpleFunc α E := fun n x ↦ (s n).comp (Prod.mk x) ⋯
f' : ℕ → β → F := fun n ↦ {x | Integrable (f x) μ}.indicator fun x ↦ SimpleFunc.setToSimpleFunc T (s' n x)
n : ℕ
x : β
⊢ {x ∈ (s' n x).range | x ≠ 0} ⊆ (s n).range
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
β : Type u_7
mβ : MeasurableSpace β
inst✝ : SFinite μ
hT : DominatedFinMeasAdditive μ T C
h'T : ∀ (s : Set (β × α)), MeasurableSet s → StronglyMeasurable fun x ↦ T (Prod.mk x ⁻¹' s)
f : β → α → E
hf : StronglyMeasurable (Function.uncurry f)
hF : CompleteSpace F
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : SeparableSpace ↑(range (Function.uncurry f) ∪ {0})
s : ℕ → SimpleFunc (β × α) E := SimpleFunc.approxOn (Function.uncurry f) ⋯ (range (Function.uncurry f) ∪ {0}) 0 ⋯
s' : ℕ → β → SimpleFunc α E := fun n x ↦ (s n).comp (Prod.mk x) ⋯
f' : ℕ → β → F := fun n ↦ {x | Integrable (f x) μ}.indicator fun x ↦ SimpleFunc.setToSimpleFunc T (s' n x)
n : ℕ
x : β
⊢ (s' n x).range ⊆ (s n).range

Tactic:
;

Goal state after:
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
β : Type u_7
mβ : MeasurableSpace β
inst✝ : SFinite μ
hT : DominatedFinMeasAdditive μ T C
h'T : ∀ (s : Set (β × α)), MeasurableSet s → StronglyMeasurable fun x ↦ T (Prod.mk x ⁻¹' s)
f : β → α → E
hf : StronglyMeasurable (Function.uncurry f)
hF : CompleteSpace F
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : SeparableSpace ↑(range (Function.uncurry f) ∪ {0})
s : ℕ → SimpleFunc (β × α) E := SimpleFunc.approxOn (Function.uncurry f) ⋯ (range (Function.uncurry f) ∪ {0}) 0 ⋯
s' : ℕ → β → SimpleFunc α E := fun n x ↦ (s n).comp (Prod.mk x) ⋯
f' : ℕ → β → F := fun n ↦ {x | Integrable (f x) μ}.indicator fun x ↦ SimpleFunc.setToSimpleFunc T (s' n x)
n : ℕ
x : β
⊢ (s' n x).range ⊆ (s n).range
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
β : Type u_7
mβ : MeasurableSpace β
inst✝ : SFinite μ
hT : DominatedFinMeasAdditive μ T C
h'T : ∀ (s : Set (β × α)), MeasurableSet s → StronglyMeasurable fun x ↦ T (Prod.mk x ⁻¹' s)
f : β → α → E
hf : StronglyMeasurable (Function.uncurry f)
hF : CompleteSpace F
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : SeparableSpace ↑(range (Function.uncurry f) ∪ {0})
s : ℕ → SimpleFunc (β × α) E := SimpleFunc.approxOn (Function.uncurry f) ⋯ (range (Function.uncurry f) ∪ {0}) 0 ⋯
s' : ℕ → β → SimpleFunc α E := fun n x ↦ (s n).comp (Prod.mk x) ⋯
f' : ℕ → β → F := fun n ↦ {x | Integrable (f x) μ}.indicator fun x ↦ SimpleFunc.setToSimpleFunc T (s' n x)
n : ℕ
x : β
y : E
⊢ y ∈ range ⇑(s' n x) → y ∈ range ⇑(s n)

Tactic:
;

Goal state after:
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
β : Type u_7
mβ : MeasurableSpace β
inst✝ : SFinite μ
hT : DominatedFinMeasAdditive μ T C
h'T : ∀ (s : Set (β × α)), MeasurableSet s → StronglyMeasurable fun x ↦ T (Prod.mk x ⁻¹' s)
f : β → α → E
hf : StronglyMeasurable (Function.uncurry f)
hF : CompleteSpace F
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : SeparableSpace ↑(range (Function.uncurry f) ∪ {0})
s : ℕ → SimpleFunc (β × α) E := SimpleFunc.approxOn (Function.uncurry f) ⋯ (range (Function.uncurry f) ∪ {0}) 0 ⋯
s' : ℕ → β → SimpleFunc α E := fun n x ↦ (s n).comp (Prod.mk x) ⋯
f' : ℕ → β → F := fun n ↦ {x | Integrable (f x) μ}.indicator fun x ↦ SimpleFunc.setToSimpleFunc T (s' n x)
n : ℕ
x : β
y : E
⊢ y ∈ range ⇑(s' n x) → y ∈ range ⇑(s n)
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
β : Type u_7
mβ : MeasurableSpace β
inst✝ : SFinite μ
hT : DominatedFinMeasAdditive μ T C
h'T : ∀ (s : Set (β × α)), MeasurableSet s → StronglyMeasurable fun x ↦ T (Prod.mk x ⁻¹' s)
f : β → α → E
hf : StronglyMeasurable (Function.uncurry f)
hF : CompleteSpace F
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : SeparableSpace ↑(range (Function.uncurry f) ∪ {0})
s : ℕ → SimpleFunc (β × α) E := SimpleFunc.approxOn (Function.uncurry f) ⋯ (range (Function.uncurry f) ∪ {0}) 0 ⋯
s' : ℕ → β → SimpleFunc α E := fun n x ↦ (s n).comp (Prod.mk x) ⋯
f' : ℕ → β → F := fun n ↦ {x | Integrable (f x) μ}.indicator fun x ↦ SimpleFunc.setToSimpleFunc T (s' n x)
n : ℕ
x : β
z : α
⊢ (s' n x) z ∈ range ⇑(s n)

Tactic:
;

Goal state after:
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
β : Type u_7
mβ : MeasurableSpace β
inst✝ : SFinite μ
hT : DominatedFinMeasAdditive μ T C
h'T : ∀ (s : Set (β × α)), MeasurableSet s → StronglyMeasurable fun x ↦ T (Prod.mk x ⁻¹' s)
f : β → α → E
hf : StronglyMeasurable (Function.uncurry f)
hF : CompleteSpace F
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : SeparableSpace ↑(range (Function.uncurry f) ∪ {0})
s : ℕ → SimpleFunc (β × α) E := SimpleFunc.approxOn (Function.uncurry f) ⋯ (range (Function.uncurry f) ∪ {0}) 0 ⋯
s' : ℕ → β → SimpleFunc α E := fun n x ↦ (s n).comp (Prod.mk x) ⋯
f' : ℕ → β → F := fun n ↦ {x | Integrable (f x) μ}.indicator fun x ↦ SimpleFunc.setToSimpleFunc T (s' n x)
n : ℕ
x : β
z : α
⊢ (s' n x) z ∈ range ⇑(s n)
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
β : Type u_7
mβ : MeasurableSpace β
inst✝ : SFinite μ
hT : DominatedFinMeasAdditive μ T C
h'T : ∀ (s : Set (β × α)), MeasurableSet s → StronglyMeasurable fun x ↦ T (Prod.mk x ⁻¹' s)
f : β → α → E
hf : StronglyMeasurable (Function.uncurry f)
hF : CompleteSpace F
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : SeparableSpace ↑(range (Function.uncurry f) ∪ {0})
s : ℕ → SimpleFunc (β × α) E := SimpleFunc.approxOn (Function.uncurry f) ⋯ (range (Function.uncurry f) ∪ {0}) 0 ⋯
s' : ℕ → β → SimpleFunc α E := fun n x ↦ (s n).comp (Prod.mk x) ⋯
f' : ℕ → β → F := fun n ↦ {x | Integrable (f x) μ}.indicator fun x ↦ SimpleFunc.setToSimpleFunc T (s' n x)
hf' : ∀ (n : ℕ), StronglyMeasurable (f' n)
x : β
hfx : Integrable (f x) μ
n : ℕ
y : α
⊢ ‖(⇑(s n) ∘ Prod.mk x) y‖ ≤ ((fun a ↦ ‖f x a‖) + fun a ↦ ‖f x a‖) y

Tactic:
;

Goal state after:
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
β : Type u_7
mβ : MeasurableSpace β
inst✝ : SFinite μ
hT : DominatedFinMeasAdditive μ T C
h'T : ∀ (s : Set (β × α)), MeasurableSet s → StronglyMeasurable fun x ↦ T (Prod.mk x ⁻¹' s)
f : β → α → E
hf : StronglyMeasurable (Function.uncurry f)
hF : CompleteSpace F
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : SeparableSpace ↑(range (Function.uncurry f) ∪ {0})
s : ℕ → SimpleFunc (β × α) E := SimpleFunc.approxOn (Function.uncurry f) ⋯ (range (Function.uncurry f) ∪ {0}) 0 ⋯
s' : ℕ → β → SimpleFunc α E := fun n x ↦ (s n).comp (Prod.mk x) ⋯
f' : ℕ → β → F := fun n ↦ {x | Integrable (f x) μ}.indicator fun x ↦ SimpleFunc.setToSimpleFunc T (s' n x)
hf' : ∀ (n : ℕ), StronglyMeasurable (f' n)
x : β
hfx : Integrable (f x) μ
n : ℕ
y : α
⊢ ‖(⇑(s n) ∘ Prod.mk x) y‖ ≤ ((fun a ↦ ‖f x a‖) + fun a ↦ ‖f x a‖) y
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝⁴ : LinearOrder R
inst✝³ : TopologicalSpace R
f : StieltjesFunction R
inst✝² : OrderTopology R
inst✝¹ : CompactIccSpace R
inst✝ : DenselyOrdered R
a b : R
hab : a < b
s : ℕ → Set R
hs : Ioc a b ⊆ ⋃ i, s i
ε : ℝ≥0
εpos : 0 < ε
h : ∑' (i : ℕ), f.length (s i) < ∞
δ : ℝ≥0 := ε / 2
δpos : 0 < ↑δ
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑δ
⊢ ContinuousWithinAt (↑f) (Ioi a) a

Tactic:
exact (f.right_continuous a).mono Ioi_subset_Ici_self

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: G : Type w
inst✝ : TopologicalSpace G
μ : Compacts G → ℝ≥0
mono'✝¹ : ∀ (K₁ K₂ : Compacts G), ↑K₁ ⊆ ↑K₂ → μ K₁ ≤ μ K₂
sup_disjoint'✝¹ : ∀ (K₁ K₂ : Compacts G), Disjoint ↑K₁ ↑K₂ → IsClosed ↑K₁ → IsClosed ↑K₂ → μ (K₁ ⊔ K₂) = μ K₁ + μ K₂
sup_le'✝¹ : ∀ (K₁ K₂ : Compacts G), μ (K₁ ⊔ K₂) ≤ μ K₁ + μ K₂
v : Compacts G → ℝ≥0
mono'✝ : ∀ (K₁ K₂ : Compacts G), ↑K₁ ⊆ ↑K₂ → v K₁ ≤ v K₂
sup_disjoint'✝ : ∀ (K₁ K₂ : Compacts G), Disjoint ↑K₁ ↑K₂ → IsClosed ↑K₁ → IsClosed ↑K₂ → v (K₁ ⊔ K₂) = v K₁ + v K₂
sup_le'✝ : ∀ (K₁ K₂ : Compacts G), v (K₁ ⊔ K₂) ≤ v K₁ + v K₂
h :
  (fun μ s ↦ ↑(μ.toFun s)) { toFun := μ, mono' := mono'✝¹, sup_disjoint' := sup_disjoint'✝¹, sup_le' := sup_le'✝¹ } =
    (fun μ s ↦ ↑(μ.toFun s)) { toFun := v, mono' := mono'✝, sup_disjoint' := sup_disjoint'✝, sup_le' := sup_le'✝ }
⊢ { toFun := μ, mono' := mono'✝¹, sup_disjoint' := sup_disjoint'✝¹, sup_le' := sup_le'✝¹ } =
    { toFun := v, mono' := mono'✝, sup_disjoint' := sup_disjoint'✝, sup_le' := sup_le'✝ }

Tactic:
;

Goal state after:
G : Type w
inst✝ : TopologicalSpace G
μ : Compacts G → ℝ≥0
mono'✝¹ : ∀ (K₁ K₂ : Compacts G), ↑K₁ ⊆ ↑K₂ → μ K₁ ≤ μ K₂
sup_disjoint'✝¹ : ∀ (K₁ K₂ : Compacts G), Disjoint ↑K₁ ↑K₂ → IsClosed ↑K₁ → IsClosed ↑K₂ → μ (K₁ ⊔ K₂) = μ K₁ + μ K₂
sup_le'✝¹ : ∀ (K₁ K₂ : Compacts G), μ (K₁ ⊔ K₂) ≤ μ K₁ + μ K₂
v : Compacts G → ℝ≥0
mono'✝ : ∀ (K₁ K₂ : Compacts G), ↑K₁ ⊆ ↑K₂ → v K₁ ≤ v K₂
sup_disjoint'✝ : ∀ (K₁ K₂ : Compacts G), Disjoint ↑K₁ ↑K₂ → IsClosed ↑K₁ → IsClosed ↑K₂ → v (K₁ ⊔ K₂) = v K₁ + v K₂
sup_le'✝ : ∀ (K₁ K₂ : Compacts G), v (K₁ ⊔ K₂) ≤ v K₁ + v K₂
h :
  (fun μ s ↦ ↑(μ.toFun s)) { toFun := μ, mono' := mono'✝¹, sup_disjoint' := sup_disjoint'✝¹, sup_le' := sup_le'✝¹ } =
    (fun μ s ↦ ↑(μ.toFun s)) { toFun := v, mono' := mono'✝, sup_disjoint' := sup_disjoint'✝, sup_le' := sup_le'✝ }
⊢ { toFun := μ, mono' := mono'✝¹, sup_disjoint' := sup_disjoint'✝¹, sup_le' := sup_le'✝¹ } =
    { toFun := v, mono' := mono'✝, sup_disjoint' := sup_disjoint'✝, sup_le' := sup_le'✝ }
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: G : Type w
inst✝ : TopologicalSpace G
μ : Compacts G → ℝ≥0
mono'✝¹ : ∀ (K₁ K₂ : Compacts G), ↑K₁ ⊆ ↑K₂ → μ K₁ ≤ μ K₂
sup_disjoint'✝¹ : ∀ (K₁ K₂ : Compacts G), Disjoint ↑K₁ ↑K₂ → IsClosed ↑K₁ → IsClosed ↑K₂ → μ (K₁ ⊔ K₂) = μ K₁ + μ K₂
sup_le'✝¹ : ∀ (K₁ K₂ : Compacts G), μ (K₁ ⊔ K₂) ≤ μ K₁ + μ K₂
v : Compacts G → ℝ≥0
mono'✝ : ∀ (K₁ K₂ : Compacts G), ↑K₁ ⊆ ↑K₂ → v K₁ ≤ v K₂
sup_disjoint'✝ : ∀ (K₁ K₂ : Compacts G), Disjoint ↑K₁ ↑K₂ → IsClosed ↑K₁ → IsClosed ↑K₂ → v (K₁ ⊔ K₂) = v K₁ + v K₂
sup_le'✝ : ∀ (K₁ K₂ : Compacts G), v (K₁ ⊔ K₂) ≤ v K₁ + v K₂
h :
  (fun μ s ↦ ↑(μ.toFun s)) { toFun := μ, mono' := mono'✝¹, sup_disjoint' := sup_disjoint'✝¹, sup_le' := sup_le'✝¹ } =
    (fun μ s ↦ ↑(μ.toFun s)) { toFun := v, mono' := mono'✝, sup_disjoint' := sup_disjoint'✝, sup_le' := sup_le'✝ }
⊢ μ = v

Tactic:
;

Goal state after:
G : Type w
inst✝ : TopologicalSpace G
μ : Compacts G → ℝ≥0
mono'✝¹ : ∀ (K₁ K₂ : Compacts G), ↑K₁ ⊆ ↑K₂ → μ K₁ ≤ μ K₂
sup_disjoint'✝¹ : ∀ (K₁ K₂ : Compacts G), Disjoint ↑K₁ ↑K₂ → IsClosed ↑K₁ → IsClosed ↑K₂ → μ (K₁ ⊔ K₂) = μ K₁ + μ K₂
sup_le'✝¹ : ∀ (K₁ K₂ : Compacts G), μ (K₁ ⊔ K₂) ≤ μ K₁ + μ K₂
v : Compacts G → ℝ≥0
mono'✝ : ∀ (K₁ K₂ : Compacts G), ↑K₁ ⊆ ↑K₂ → v K₁ ≤ v K₂
sup_disjoint'✝ : ∀ (K₁ K₂ : Compacts G), Disjoint ↑K₁ ↑K₂ → IsClosed ↑K₁ → IsClosed ↑K₂ → v (K₁ ⊔ K₂) = v K₁ + v K₂
sup_le'✝ : ∀ (K₁ K₂ : Compacts G), v (K₁ ⊔ K₂) ≤ v K₁ + v K₂
h :
  (fun μ s ↦ ↑(μ.toFun s)) { toFun := μ, mono' := mono'✝¹, sup_disjoint' := sup_disjoint'✝¹, sup_le' := sup_le'✝¹ } =
    (fun μ s ↦ ↑(μ.toFun s)) { toFun := v, mono' := mono'✝, sup_disjoint' := sup_disjoint'✝, sup_le' := sup_le'✝ }
⊢ μ = v
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: G : Type w
inst✝ : TopologicalSpace G
μ : Compacts G → ℝ≥0
mono'✝¹ : ∀ (K₁ K₂ : Compacts G), ↑K₁ ⊆ ↑K₂ → μ K₁ ≤ μ K₂
sup_disjoint'✝¹ : ∀ (K₁ K₂ : Compacts G), Disjoint ↑K₁ ↑K₂ → IsClosed ↑K₁ → IsClosed ↑K₂ → μ (K₁ ⊔ K₂) = μ K₁ + μ K₂
sup_le'✝¹ : ∀ (K₁ K₂ : Compacts G), μ (K₁ ⊔ K₂) ≤ μ K₁ + μ K₂
v : Compacts G → ℝ≥0
mono'✝ : ∀ (K₁ K₂ : Compacts G), ↑K₁ ⊆ ↑K₂ → v K₁ ≤ v K₂
sup_disjoint'✝ : ∀ (K₁ K₂ : Compacts G), Disjoint ↑K₁ ↑K₂ → IsClosed ↑K₁ → IsClosed ↑K₂ → v (K₁ ⊔ K₂) = v K₁ + v K₂
sup_le'✝ : ∀ (K₁ K₂ : Compacts G), v (K₁ ⊔ K₂) ≤ v K₁ + v K₂
h :
  (fun μ s ↦ ↑(μ.toFun s)) { toFun := μ, mono' := mono'✝¹, sup_disjoint' := sup_disjoint'✝¹, sup_le' := sup_le'✝¹ } =
    (fun μ s ↦ ↑(μ.toFun s)) { toFun := v, mono' := mono'✝, sup_disjoint' := sup_disjoint'✝, sup_le' := sup_le'✝ }
s : Compacts G
⊢ μ s = v s

Tactic:
;

Goal state after:
G : Type w
inst✝ : TopologicalSpace G
μ : Compacts G → ℝ≥0
mono'✝¹ : ∀ (K₁ K₂ : Compacts G), ↑K₁ ⊆ ↑K₂ → μ K₁ ≤ μ K₂
sup_disjoint'✝¹ : ∀ (K₁ K₂ : Compacts G), Disjoint ↑K₁ ↑K₂ → IsClosed ↑K₁ → IsClosed ↑K₂ → μ (K₁ ⊔ K₂) = μ K₁ + μ K₂
sup_le'✝¹ : ∀ (K₁ K₂ : Compacts G), μ (K₁ ⊔ K₂) ≤ μ K₁ + μ K₂
v : Compacts G → ℝ≥0
mono'✝ : ∀ (K₁ K₂ : Compacts G), ↑K₁ ⊆ ↑K₂ → v K₁ ≤ v K₂
sup_disjoint'✝ : ∀ (K₁ K₂ : Compacts G), Disjoint ↑K₁ ↑K₂ → IsClosed ↑K₁ → IsClosed ↑K₂ → v (K₁ ⊔ K₂) = v K₁ + v K₂
sup_le'✝ : ∀ (K₁ K₂ : Compacts G), v (K₁ ⊔ K₂) ≤ v K₁ + v K₂
h :
  (fun μ s ↦ ↑(μ.toFun s)) { toFun := μ, mono' := mono'✝¹, sup_disjoint' := sup_disjoint'✝¹, sup_le' := sup_le'✝¹ } =
    (fun μ s ↦ ↑(μ.toFun s)) { toFun := v, mono' := mono'✝, sup_disjoint' := sup_disjoint'✝, sup_le' := sup_le'✝ }
s : Compacts G
⊢ μ s = v s
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: G : Type w
inst✝ : TopologicalSpace G
μ : Content G
U : Opens G
hU : μ.innerContent U ≠ ∞
ε : ℝ≥0
hε : ε ≠ 0
h'ε : ↑ε ≠ 0
h : ↑ε < μ.innerContent U
h₂ : μ.innerContent U - ↑ε < μ.innerContent U
| μ.innerContent U

Tactic:
;

Goal state after:
G : Type w
inst✝ : TopologicalSpace G
μ : Content G
U : Opens G
hU : μ.innerContent U ≠ ∞
ε : ℝ≥0
hε : ε ≠ 0
h'ε : ↑ε ≠ 0
h : ↑ε < μ.innerContent U
h₂ : μ.innerContent U - ↑ε < μ.innerContent U
| μ.innerContent U
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case pos
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
t : Set X
μ : Measure X
ht_eq : ∀ᵐ (x : X) ∂μ, x ∈ t → f x = 0
hf : AEStronglyMeasurable f (μ.restrict t)
⊢ ∫ (x : X) in t, f x ∂μ = 0

pp: case neg
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
t : Set X
μ : Measure X
ht_eq : ∀ᵐ (x : X) ∂μ, x ∈ t → f x = 0
hf : ¬AEStronglyMeasurable f (μ.restrict t)
⊢ ∫ (x : X) in t, f x ∂μ = 0

Tactic:
;

Goal state after:
case pos
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
t : Set X
μ : Measure X
ht_eq : ∀ᵐ (x : X) ∂μ, x ∈ t → f x = 0
hf : AEStronglyMeasurable f (μ.restrict t)
⊢ ∫ (x : X) in t, f x ∂μ = 0

case neg
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
t : Set X
μ : Measure X
ht_eq : ∀ᵐ (x : X) ∂μ, x ∈ t → f x = 0
hf : ¬AEStronglyMeasurable f (μ.restrict t)
⊢ ∫ (x : X) in t, f x ∂μ = 0
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: case inr
G : Type w
inst✝ : TopologicalSpace G
μ : Content G
U✝ : Opens G
hU : μ.innerContent U✝ ≠ ∞
ε : ℝ≥0
hε : ε ≠ 0
h'ε : ↑ε ≠ 0
h : ↑ε < μ.innerContent U✝
U : Compacts G
h1U : ↑U ⊆ ↑U✝
h2U : μ.innerContent U✝ - ↑ε < μ U
⊢ ∃ K, ↑K ⊆ ↑U✝ ∧ μ.innerContent U✝ ≤ μ K + ↑ε

Tactic:
;

Goal state after:
case inr
G : Type w
inst✝ : TopologicalSpace G
μ : Content G
U✝ : Opens G
hU : μ.innerContent U✝ ≠ ∞
ε : ℝ≥0
hε : ε ≠ 0
h'ε : ↑ε ≠ 0
h : ↑ε < μ.innerContent U✝
U : Compacts G
h1U : ↑U ⊆ ↑U✝
h2U : μ.innerContent U✝ - ↑ε < μ U
⊢ ∃ K, ↑K ⊆ ↑U✝ ∧ μ.innerContent U✝ ≤ μ K + ↑ε
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: case inr
G : Type w
inst✝ : TopologicalSpace G
μ : Content G
U✝ : Opens G
hU : μ.innerContent U✝ ≠ ∞
ε : ℝ≥0
hε : ε ≠ 0
h'ε : ↑ε ≠ 0
h : ↑ε < μ.innerContent U✝
U : Compacts G
h1U : ↑U ⊆ ↑U✝
h2U : μ.innerContent U✝ - ↑ε < μ U
⊢ μ.innerContent U✝ - ↑ε ≤ μ U

Tactic:
;

Goal state after:
case inr
G : Type w
inst✝ : TopologicalSpace G
μ : Content G
U✝ : Opens G
hU : μ.innerContent U✝ ≠ ∞
ε : ℝ≥0
hε : ε ≠ 0
h'ε : ↑ε ≠ 0
h : ↑ε < μ.innerContent U✝
U : Compacts G
h1U : ↑U ⊆ ↑U✝
h2U : μ.innerContent U✝ - ↑ε < μ U
⊢ μ.innerContent U✝ - ↑ε ≤ μ U
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
t : Set X
μ : Measure X
hU : ∀ x ∈ t, f x = 0
⊢ ∫ (x : X) in t, f x ∂μ = 0

Tactic:
;

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
t : Set X
μ : Measure X
hU : ∀ x ∈ t, f x = 0
⊢ ∫ (x : X) in t, f x ∂μ = 0
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s t : Set X
μ : Measure 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}
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
⊢ MeasurableSet k

Tactic:
;

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s t : Set X
μ : Measure 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}
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
⊢ MeasurableSet k
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
α : Type u_3
inst✝⁴ : Group G
inst✝³ : MulAction G α
inst✝² : MeasurableSpace α
s : Set α
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : Countable G
h_meas : NullMeasurableSet s μ
h_ae_disjoint : ∀ (g : G), g ≠ 1 → AEDisjoint μ (g • s) s
h_qmp : ∀ (g : G), QuasiMeasurePreserving (fun x ↦ g • x) μ μ
h_measure_univ_le : μ univ ≤ ∑' (g : G), μ (g • s)
aedisjoint : Pairwise (AEDisjoint μ on fun g ↦ g • s)
g : G
⊢ NullMeasurableSet ((fun x ↦ g⁻¹ • x) ⁻¹' s) μ

Tactic:
;

Goal state after:
G : Type u_1
α : Type u_3
inst✝⁴ : Group G
inst✝³ : MulAction G α
inst✝² : MeasurableSpace α
s : Set α
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : Countable G
h_meas : NullMeasurableSet s μ
h_ae_disjoint : ∀ (g : G), g ≠ 1 → AEDisjoint μ (g • s) s
h_qmp : ∀ (g : G), QuasiMeasurePreserving (fun x ↦ g • x) μ μ
h_measure_univ_le : μ univ ≤ ∑' (g : G), μ (g • s)
aedisjoint : Pairwise (AEDisjoint μ on fun g ↦ g • s)
g : G
⊢ NullMeasurableSet ((fun x ↦ g⁻¹ • x) ⁻¹' s) μ
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s t : Set X
μ : Measure X
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
⊢ (fun x ↦ 0) =ᵐ[μ.restrict t] f

Tactic:
;

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s t : Set X
μ : Measure X
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
⊢ (fun x ↦ 0) =ᵐ[μ.restrict t] f
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s t : Set X
μ : Measure X
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
⊢ f =ᵐ[μ.restrict t] fun x ↦ 0

Tactic:
;

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s t : Set X
μ : Measure X
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
⊢ f =ᵐ[μ.restrict t] fun x ↦ 0
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
α : Type u_3
inst✝⁴ : Group G
inst✝³ : MulAction G α
inst✝² : MeasurableSpace α
s : Set α
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : Countable G
h_ae_disjoint : ∀ (g : G), g ≠ 1 → AEDisjoint μ (g • s) s
h_qmp : ∀ (g : G), QuasiMeasurePreserving (fun x ↦ g • x) μ μ
h_measure_univ_le : μ univ ≤ ∑' (g : G), μ (g • s)
aedisjoint : Pairwise (AEDisjoint μ on fun g ↦ g • s)
h_meas : ∀ (g : G), NullMeasurableSet (g • s) μ
⊢ NullMeasurableSet (⋃ g, g • s) μ

Tactic:
;

Goal state after:
G : Type u_1
α : Type u_3
inst✝⁴ : Group G
inst✝³ : MulAction G α
inst✝² : MeasurableSpace α
s : Set α
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : Countable G
h_ae_disjoint : ∀ (g : G), g ≠ 1 → AEDisjoint μ (g • s) s
h_qmp : ∀ (g : G), QuasiMeasurePreserving (fun x ↦ g • x) μ μ
h_measure_univ_le : μ univ ≤ ∑' (g : G), μ (g • s)
aedisjoint : Pairwise (AEDisjoint μ on fun g ↦ g • s)
h_meas : ∀ (g : G), NullMeasurableSet (g • s) μ
⊢ NullMeasurableSet (⋃ g, g • s) μ