mathlib-initiative/mathlib-tactics huggingface dataset:
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
a' : R
ha' : ↑f a' - ↑f a < ↑δ
aa' : a < a'
this : Nonempty R
i : ℕ
p q' : R
spq : s i \ botSet ⊆ Ioc p q'
hq' : ofReal (↑f q' - ↑f p) < f.length (s i) + ↑(ε' i)
A : Icc a' b ∩ s i ⊆ Ioc p q'
htq' : ¬IsTop q'
⊢ (𝓝[>] q').NeBot

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
α : Type u_3
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : MeasurableSpace α
s : Set α
μ : Measure α
h_meas : NullMeasurableSet s μ
h_exists : ∀ (x : α), ∃! g, g • x ∈ s
a b : G
hab : a ≠ b
x : α
hxa : x ∈ (fun g ↦ g • s) a
hxb : x ∈ (fun g ↦ g • s) b
⊢ False

Tactic:
by

Goal state after:
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 (g • s) μ

Tactic:
by

Goal state after:
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
a' : R
ha' : ↑f a' - ↑f a < ↑δ
aa' : a < a'
this✝ : Nonempty R
i : ℕ
p q' : R
spq : s i \ botSet ⊆ Ioc p q'
hq' : ofReal (↑f q' - ↑f p) < f.length (s i) + ↑(ε' i)
A : Icc a' b ∩ s i ⊆ Ioc p q'
htq' : ¬IsTop q'
this : (𝓝[>] q').NeBot
⊢ ContinuousWithinAt (fun r ↦ ofReal (↑f r - ↑f p)) (Ioi q') q'

Tactic:
by

Goal state after:
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
⊢ ∫ (x : X) in s ∪ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ

Tactic:
by

Goal state after:
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
⊢ IntegrableOn f t μ

Tactic:
by

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

Goal state before:
pp: G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
⊢ μ.innerContent (⨆ i, U i) ≤ ∑' (i : ℕ), μ.innerContent (U i)

Tactic:
by

Goal state after:
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 {a | ∃ g, g • a ∈ s} μ

Tactic:
by

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

Goal state before:
pp: G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Set G
hU : ∀ (i : ℕ), IsOpen (U i)
⊢ μ.innerContent { carrier := ⋃ i, U i, is_open' := ⋯ } ≤ ∑' (i : ℕ), μ.innerContent { carrier := U i, is_open' := ⋯ }

Tactic:
by

Goal state after:
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
a' : R
ha' : ↑f a' - ↑f a < ↑δ
aa' : a < a'
this : Nonempty R
⊢ ∀ (i : ℕ), ∃ p, Icc a' b ∩ s i ⊆ Iotop p.1 p.2 ∧ ofReal (↑f p.2 - ↑f p.1) < f.length (s i) + ↑(ε' i)

Tactic:
by

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

Goal state before:
pp: G : Type w
inst✝ : TopologicalSpace G
μ : Content G
f : G ≃ₜ G
h : ∀ ⦃K : Compacts G⦄, μ (Compacts.map ⇑f ⋯ K) = μ K
U : Opens G
⊢ μ.innerContent ((Opens.comap ↑f) U) = μ.innerContent U

Tactic:
by

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

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : Group G
inst✝ : SeparatelyContinuousMul G
h : ∀ (g : G) {K : Compacts G}, μ (Compacts.map (fun x ↦ g * x) ⋯ K) = μ K
g : G
U : Opens G
⊢ μ.innerContent ((Opens.comap ↑(Homeomorph.mulLeft g)) U) = μ.innerContent U

Tactic:
by

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

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
h3 : ∀ (g : G) {K : Compacts G}, μ (Compacts.map (fun x ↦ g * x) ⋯ K) = μ K
K : Compacts G
hK : μ K ≠ 0
U : Opens G
hU : (↑U).Nonempty
⊢ (interior ↑U).Nonempty

Tactic:
by

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

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
h3 : ∀ (g : G) {K : Compacts G}, μ (Compacts.map (fun x ↦ g * x) ⋯ K) = μ K
K : Compacts G
hK : μ K ≠ 0
U : Opens G
hU : (↑U).Nonempty
this✝ : (interior ↑U).Nonempty
s : Finset G
hs : K.carrier ⊆ ⋃ g ∈ s, (fun x ↦ g * x) ⁻¹' ↑U
this : μ K ≤ ↑s.card * μ.innerContent U
⊢ 0 < μ.innerContent U

Tactic:
by

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

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
h3 : ∀ (g : G) {K : Compacts G}, μ (Compacts.map (fun x ↦ g * x) ⋯ K) = μ K
K : Compacts G
hK : μ K ≠ 0
U : Opens G
hU : (↑U).Nonempty
this : (interior ↑U).Nonempty
s : Finset G
hs : K.carrier ⊆ ⋃ g ∈ s, (fun x ↦ g * x) ⁻¹' ↑U
⊢ ↑K ⊆ ↑(⨆ g ∈ s, (Opens.comap ↑(Homeomorph.mulLeft g)) U)

Tactic:
by

Goal state after:
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
a' : R
ha' : ↑f a' - ↑f a < ↑δ
aa' : a < a'
this : Nonempty R
g : ℕ → R × R
hg : ∀ (i : ℕ), Icc a' b ∩ s i ⊆ Iotop (g i).1 (g i).2 ∧ ofReal (↑f (g i).2 - ↑f (g i).1) < f.length (s i) + ↑(ε' i)
⊢ Icc a' b ∩ Ioc a b ⊆ Icc a' b ∩ ⋃ i, s i

Tactic:
by

Goal state after:
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)
⊢ ∀ᵐ (x : α) ∂μ, ∃ g, g • x ∈ s

Tactic:
by

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

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
h3 : ∀ (g : G) {K : Compacts G}, μ (Compacts.map (fun x ↦ g * x) ⋯ K) = μ K
K : Compacts G
hK : μ K ≠ 0
U : Opens G
hU : (↑U).Nonempty
⊢ 0 < μ.innerContent U

Tactic:
by

Goal state after:
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
ht : IntegrableOn f t μ
H : IntegrableOn f (s ∪ t) μ
f' : X → E := AEStronglyMeasurable.mk f ⋯
⊢ ∫ (x : X) in s ∪ t, f' x ∂μ = ∫ (x : X) in s, f' x ∂μ

Tactic:
by

Goal state after:
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
a' : R
ha' : ↑f a' - ↑f a < ↑δ
aa' : a < a'
this : Nonempty R
g : ℕ → R × R
hg : ∀ (i : ℕ), Icc a' b ∩ s i ⊆ Iotop (g i).1 (g i).2 ∧ ofReal (↑f (g i).2 - ↑f (g i).1) < f.length (s i) + ↑(ε' i)
I_subset : Icc a' b ⊆ ⋃ i, Iotop (g i).1 (g i).2
⊢ ofReal (↑f b - ↑f a) = ofReal (↑f b - ↑f a' + (↑f a' - ↑f a))

Tactic:
by

Goal state after:
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
⊢ ∫ (x : X) in s ∪ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ

Tactic:
by

Goal state after:
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 : Countable G
inst✝ : SMulInvariantMeasure G α μ
hμ : μ ≠ 0
h : IsFundamentalDomain G s μ
⊢ μ s ≠ 0

Tactic:
by

Goal state after:
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
a' : R
ha' : ↑f a' - ↑f a < ↑δ
aa' : a < a'
this : Nonempty R
g : ℕ → R × R
hg : ∀ (i : ℕ), Icc a' b ∩ s i ⊆ Iotop (g i).1 (g i).2 ∧ ofReal (↑f (g i).2 - ↑f (g i).1) < f.length (s i) + ↑(ε' i)
I_subset : Icc a' b ⊆ ⋃ i, Iotop (g i).1 (g i).2
⊢ ofReal ↑δ ≤ ↑δ

Tactic:
by

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

Goal state before:
pp: G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
K : Compacts G
⊢ μ K ≤ μ.outerMeasure ↑K

Tactic:
by

Goal state after:
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
hts : s ⊆ t
h't : ∀ᵐ (x : X) ∂μ, x ∈ t \ s → f x = 0
haux : StronglyMeasurable f
h'aux : IntegrableOn f t μ
k : Set X := f ⁻¹' {0}
⊢ MeasurableSet k

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
H : Type u_2
α : Type u_3
β : Type u_4
inst✝⁵ : Group G
inst✝⁴ : Group H
inst✝³ : MulAction G α
inst✝² : MeasurableSpace α
inst✝¹ : MulAction H β
inst✝ : MeasurableSpace β
s : Set α
μ : Measure α
ν : Measure β
h : IsFundamentalDomain G s μ
f : β → α
hf : QuasiMeasurePreserving f ν μ
e : G → H
he : Bijective e
hef : ∀ (g : G), Semiconj f (fun x ↦ e g • x) fun x ↦ g • x
x : β
x✝ : ∃ g, g • f x ∈ s
g : G
hg : g • f x ∈ s
⊢ e g • x ∈ f ⁻¹' s

Tactic:
by

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

Goal state before:
pp: G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : Opens G
hU : μ.innerContent U ≠ ∞
ε : ℝ≥0
hε : ε ≠ 0
K : Compacts G
h1K : ↑K ⊆ ↑U
h2K : μ.innerContent U ≤ μ K + ↑ε
⊢ μ.innerContent U ≤ μ.outerMeasure ↑K + ↑ε

Tactic:
by

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

Goal state before:
pp: G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : Opens G
hU : μ.outerMeasure ↑U ≠ ∞
ε : ℝ≥0
hε : ε ≠ 0
⊢ ∃ K, ↑K ⊆ ↑U ∧ μ.outerMeasure ↑U ≤ μ.outerMeasure ↑K + ↑ε

Tactic:
by

Goal state after:
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
a' : R
ha' : ↑f a' - ↑f a < ↑δ
aa' : a < a'
this : Nonempty R
g : ℕ → R × R
hg : ∀ (i : ℕ), Icc a' b ∩ s i ⊆ Iotop (g i).1 (g i).2 ∧ ofReal (↑f (g i).2 - ↑f (g i).1) < f.length (s i) + ↑(ε' i)
I_subset : Icc a' b ⊆ ⋃ i, Iotop (g i).1 (g i).2
⊢ ∑' (i : ℕ), (f.length (s i) + ↑(ε' i)) + ↑δ = ∑' (i : ℕ), f.length (s i) + ∑' (i : ℕ), ↑(ε' i) + ↑δ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
H : Type u_2
α : Type u_3
β : Type u_4
inst✝⁵ : Group G
inst✝⁴ : Group H
inst✝³ : MulAction G α
inst✝² : MeasurableSpace α
inst✝¹ : MulAction H β
inst✝ : MeasurableSpace β
s : Set α
μ : Measure α
ν : Measure β
h : IsFundamentalDomain G s μ
f : β → α
hf : QuasiMeasurePreserving f ν μ
a b : H
hab : a ≠ b
e : G ≃ H
hef : ∀ (g : G), Semiconj f (fun x ↦ e g • x) fun x ↦ g • x
⊢ (e.symm a⁻¹)⁻¹ ≠ (e.symm b⁻¹)⁻¹

Tactic:
by

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

Goal state before:
pp: G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
A : Set G
hA : μ.outerMeasure A ≠ ∞
ε : ℝ≥0
hε : ε ≠ 0
⊢ ∃ U, A ⊆ ↑U ∧ μ.outerMeasure ↑U ≤ μ.outerMeasure A + ↑ε

Tactic:
by

Goal state after:
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
hts : s ⊆ t
h't : ∀ᵐ (x : X) ∂μ, x ∈ t \ s → f x = 0
haux : StronglyMeasurable f
h'aux : IntegrableOn f t μ
k : Set X := f ⁻¹' {0}
hk : MeasurableSet k
⊢ ∫ (x : X) in t, f x ∂μ = ∫ (x : X) in t ∩ k, f x ∂μ + ∫ (x : X) in t \ k, f x ∂μ

Tactic:
by

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

Goal state before:
pp: G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
f : G ≃ₜ G
h : ∀ ⦃K : Compacts G⦄, μ (Compacts.map ⇑f ⋯ K) = μ K
A : Set G
⊢ μ.outerMeasure (⇑f ⁻¹' A) = μ.outerMeasure A

Tactic:
by

Goal state after:
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
a' : R
ha' : ↑f a' - ↑f a < ↑δ
aa' : a < a'
this : Nonempty R
g : ℕ → R × R
hg : ∀ (i : ℕ), Icc a' b ∩ s i ⊆ Iotop (g i).1 (g i).2 ∧ ofReal (↑f (g i).2 - ↑f (g i).1) < f.length (s i) + ↑(ε' i)
I_subset : Icc a' b ⊆ ⋃ i, Iotop (g i).1 (g i).2
⊢ ∑' (i : ℕ), f.length (s i) + ↑δ + ↑δ = ∑' (i : ℕ), f.length (s i) + ↑ε

Tactic:
by

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

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
inst✝ : WeaklyLocallyCompactSpace G
K : Set G
hK : IsCompact K
F : Set G
h1F : IsCompact F
h2F : K ⊆ interior F
⊢ μ.outerMeasure (interior F) ≤ μ { carrier := F, isCompact' := h1F }

Tactic:
by

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

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
inst✝ : WeaklyLocallyCompactSpace G
K : Set G
hK : IsCompact K
⊢ μ.outerMeasure K < ∞

Tactic:
by

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

Goal state before:
pp: G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : R1Space G
inst✝¹ : Group G
inst✝ : SeparatelyContinuousMul G
h : ∀ (g : G) {K : Compacts G}, μ (Compacts.map (fun x ↦ g * x) ⋯ K) = μ K
g : G
A : Set G
⊢ μ.outerMeasure ((fun x ↦ g * x) ⁻¹' A) = μ.outerMeasure A

Tactic:
by

Goal state after:
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
⊢ f.outer (Ioc a b) = ofReal (↑f b - ↑f a)

Tactic:
by

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

Goal state before:
pp: G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
A : Set G
⊢ MeasurableSet A ↔ ∀ (U : Opens G), μ.outerMeasure (↑U ∩ A) + μ.outerMeasure (↑U \ A) ≤ μ.outerMeasure ↑U

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
H : Type u_2
α : Type u_3
β : Type u_4
inst✝⁵ : Group G
inst✝⁴ : Group H
inst✝³ : MulAction G α
inst✝² : MeasurableSpace α
inst✝¹ : MulAction H β
inst✝ : MeasurableSpace β
s : Set α
μ : Measure α
ν : Measure β
h : IsFundamentalDomain G s μ
f : β → α
hf : QuasiMeasurePreserving f ν μ
e : G → H
he : Bijective e
hef : ∀ (g : G), Semiconj f (fun x ↦ e g • x) fun x ↦ g • x
a b : H
hab : a ≠ b
⊢ (AEDisjoint ν on fun g ↦ g • f ⁻¹' s) a b

Tactic:
by

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

Goal state before:
pp: G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : R1Space G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
h3 : ∀ (g : G) {K : Compacts G}, μ (Compacts.map (fun x ↦ g * x) ⋯ K) = μ K
K : Compacts G
hK : μ K ≠ 0
U : Set G
h1U : IsOpen U
h2U : U.Nonempty
⊢ 0 < μ.outerMeasure U

Tactic:
by

Goal state after:
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
hts : s ⊆ t
h't : ∀ᵐ (x : X) ∂μ, x ∈ t \ s → f x = 0
haux : StronglyMeasurable f
h'aux : IntegrableOn f t μ
k : Set X := f ⁻¹' {0}
hk : MeasurableSet k
⊢ ∫ (x : X) in t ∩ k, f x ∂μ + ∫ (x : X) in t \ k, f x ∂μ = ∫ (x : X) in t \ k, f x ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
H : Type u_2
α : Type u_3
β : Type u_4
inst✝⁵ : Group G
inst✝⁴ : Group H
inst✝³ : MulAction G α
inst✝² : MeasurableSpace α
inst✝¹ : MulAction H β
inst✝ : MeasurableSpace β
s : Set α
μ : Measure α
ν : Measure β
h : IsFundamentalDomain G s μ
f : α ≃ β
hf : QuasiMeasurePreserving (⇑f.symm) ν μ
e : H ≃ G
hef : ∀ (g : H), Semiconj (⇑f) (fun x ↦ e g • x) fun x ↦ g • x
⊢ IsFundamentalDomain H (⇑f '' s) ν

Tactic:
by

Goal state after:
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 : MeasurableConstSMul G α
inst✝ : SMulInvariantMeasure G α μ
h : IsFundamentalDomain G s μ
g g' : G
⊢ (fun g' ↦ g * g' * g⁻¹) ((fun g' ↦ g⁻¹ * g' * g) g') = g'

Tactic:
by

Goal state after:
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 : MeasurableConstSMul G α
inst✝ : SMulInvariantMeasure G α μ
h : IsFundamentalDomain G s μ
g g' : G
⊢ (fun g' ↦ g⁻¹ * g' * g) ((fun g' ↦ g * g' * g⁻¹) g') = g'

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝¹ : LinearOrder R
inst✝ : TopologicalSpace R
f : StieltjesFunction R
c : R
⊢ MeasurableSet (Ioi c)

Tactic:
by

Goal state after:
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 : MeasurableConstSMul G α
inst✝ : SMulInvariantMeasure G α μ
h : IsFundamentalDomain G s μ
g g' : G
x : α
⊢ (MulAction.toPerm g)
      ((fun x ↦
          { toFun := fun g' ↦ g⁻¹ * g' * g, invFun := fun g' ↦ g * g' * g⁻¹, left_inv := ⋯, right_inv := ⋯ } g' • x)
        x) =
    (fun x ↦ g' • x) ((MulAction.toPerm g) x)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
α : Type u_3
inst✝⁵ : Group G
inst✝⁴ : MulAction G α
inst✝³ : MeasurableSpace α
s : Set α
μ : Measure α
inst✝² : MeasurableConstSMul G α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : Countable G
ν : Measure α
h : IsFundamentalDomain G s μ
hν : ν ≪ μ
⊢ (sum fun g ↦ ν.restrict (g • s)) = ν

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
α : Type u_3
inst✝⁵ : Group G
inst✝⁴ : MulAction G α
inst✝³ : MeasurableSpace α
s : Set α
μ : Measure α
inst✝² : MeasurableConstSMul G α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : Countable G
ν : Measure α
h : IsFundamentalDomain G s μ
hν : ν ≪ μ
f : α → ℝ≥0∞
⊢ ∫⁻ (x : α), f x ∂ν = ∑' (g : G), ∫⁻ (x : α) in g • s, f x ∂ν

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
α : Type u_3
inst✝⁵ : Group G
inst✝⁴ : MulAction G α
inst✝³ : MeasurableSpace α
s : Set α
μ : Measure α
inst✝² : MeasurableConstSMul G α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : Countable G
h : IsFundamentalDomain G s μ
f : α → ℝ≥0∞
t : Set α
⊢ ∑' (g : G), ∫⁻ (x : α) in g • s, f x ∂μ.restrict t = ∑' (g : G), ∫⁻ (x : α) in t ∩ g • s, f x ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
α : Type u_3
inst✝⁵ : Group G
inst✝⁴ : MulAction G α
inst✝³ : MeasurableSpace α
s : Set α
μ : Measure α
inst✝² : MeasurableConstSMul G α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : Countable G
h : IsFundamentalDomain G s μ
f : α → ℝ≥0∞
t : Set α
⊢ ∑' (g : G), ∫⁻ (x : α) in t ∩ g⁻¹ • s, f x ∂μ = ∑' (g : G), ∫⁻ (x : α) in g⁻¹ • (g • t ∩ s), f x ∂μ

Tactic:
by

Goal state after:
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
hts : s ⊆ t
h't : ∀ᵐ (x : X) ∂μ, x ∈ t \ s → f x = 0
haux : StronglyMeasurable f
h'aux : IntegrableOn f t μ
k : Set X := f ⁻¹' {0}
hk : MeasurableSet k
⊢ ∫ (x : X) in t \ k, f x ∂μ = ∫ (x : X) in s \ k, f x ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
α : Type u_3
inst✝⁵ : Group G
inst✝⁴ : MulAction G α
inst✝³ : MeasurableSpace α
s : Set α
μ : Measure α
inst✝² : MeasurableConstSMul G α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : Countable G
ν : Measure α
h : IsFundamentalDomain G s μ
hν : ν ≪ μ
t : Set α
⊢ ν t = ∑' (g : G), ν (t ∩ g • s)

Tactic:
by

Goal state after: