mathlib-initiative/mathlib-tactics huggingface dataset:
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.symm ⁻¹' s) ν

Tactic:
]

Goal state after:
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.symm ⁻¹' s) ν
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
t : Set R
this : Nonempty R
a b : R
h : t \ botSet ⊆ Ioc a b
⊢ f.length (Ioc a b ∩ Ioi c) + f.length (Ioc a b \ Ioi c) ≤ ofReal (↑f b - ↑f a)

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : LinearOrder R
inst✝ : TopologicalSpace R
f : StieltjesFunction R
c : R
t : Set R
this : Nonempty R
a b : R
h : t \ botSet ⊆ Ioc a b
⊢ f.length (Ioc a b ∩ Ioi c) + f.length (Ioc a b \ Ioi c) ≤ ofReal (↑f b - ↑f a)
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
⊢ ∀ (U_1 : Opens G), μ.outerMeasure (↑U_1 ∩ U) + μ.outerMeasure (↑U_1 \ U) ≤ μ.outerMeasure ↑U_1

Tactic:
]

Goal state after:
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
⊢ ∀ (U_1 : Opens G), μ.outerMeasure (↑U_1 ∩ U) + μ.outerMeasure (↑U_1 \ U) ≤ μ.outerMeasure ↑U_1
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
⊢ μ.innerContent { carrier := ↑U' ∩ U, is_open' := ⋯ } + μ.outerMeasure (↑U' \ U) ≤ μ.outerMeasure ↑U'

Tactic:
]

Goal state after:
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
⊢ μ.innerContent { carrier := ↑U' ∩ U, is_open' := ⋯ } + μ.outerMeasure (↑U' \ U) ≤ μ.outerMeasure ↑U'
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
⊢ (⨆ x, μ ↑x) + μ.outerMeasure (↑U' \ U) ≤ μ.outerMeasure ↑U'

Tactic:
]

Goal state after:
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
⊢ (⨆ x, μ ↑x) + μ.outerMeasure (↑U' \ U) ≤ μ.outerMeasure ↑U'
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
⊢ ⨆ i, μ ↑i + μ.outerMeasure (↑U' \ U) ≤ μ.outerMeasure ↑U'

Tactic:
]

Goal state after:
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
⊢ ⨆ i, μ ↑i + μ.outerMeasure (↑U' \ U) ≤ μ.outerMeasure ↑U'
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
g : G
x : α
⊢ (fun x ↦ g • x) x = (fun x ↦ g • x) x

Tactic:
]

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
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
hL : ↑L ⊆ ↑U' ∩ U
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
⊢ μ L' + μ.outerMeasure (↑U' \ U) ≤ μ.outerMeasure ↑U'

Tactic:
]

Goal state after:
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
hL : ↑L ⊆ ↑U' ∩ U
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
⊢ μ L' + μ.outerMeasure (↑U' \ U) ≤ μ.outerMeasure ↑U'
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
hL'U : ↑L' ⊆ U
hL'U' : ↑L' ⊆ ↑U'
this : ↑U' \ U ⊆ ↑U' \ ↑L'
⊢ μ L' + μ.outerMeasure (↑U' \ ↑L') ≤ μ.outerMeasure ↑U'

Tactic:
]

Goal state after:
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
hL'U : ↑L' ⊆ U
hL'U' : ↑L' ⊆ ↑U'
this : ↑U' \ U ⊆ ↑U' \ ↑L'
⊢ μ L' + μ.outerMeasure (↑U' \ ↑L') ≤ μ.outerMeasure ↑U'
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝⁶ : LinearOrder R
inst✝⁵ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁴ : OrderTopology R
inst✝³ : CompactIccSpace R
inst✝² : MeasurableSpace R
inst✝¹ : BorelSpace R
inst✝ : DenselyOrdered R
s : Set R
⊢ ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), f.outer t ≤ f.outer s

Tactic:
]

Goal state after:
R : Type u_1
inst✝⁶ : LinearOrder R
inst✝⁵ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁴ : OrderTopology R
inst✝³ : CompactIccSpace R
inst✝² : MeasurableSpace R
inst✝¹ : BorelSpace R
inst✝ : DenselyOrdered R
s : Set R
⊢ ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), f.outer t ≤ f.outer s
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
⊢ ν = ν

Tactic:
]

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
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
hL'U : ↑L' ⊆ U
hL'U' : ↑L' ⊆ ↑U'
this : ↑U' \ U ⊆ ↑U' \ ↑L'
⊢ μ L' + μ.innerContent { carrier := ↑U' \ ↑L', is_open' := ⋯ } ≤ μ.outerMeasure ↑U'

Tactic:
]

Goal state after:
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
hL'U : ↑L' ⊆ U
hL'U' : ↑L' ⊆ ↑U'
this : ↑U' \ U ⊆ ↑U' \ ↑L'
⊢ μ L' + μ.innerContent { carrier := ↑U' \ ↑L', is_open' := ⋯ } ≤ μ.outerMeasure ↑U'
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 = ∫⁻ (a : α), f a ∂ν

Tactic:
]

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
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
hL'U : ↑L' ⊆ U
hL'U' : ↑L' ⊆ ↑U'
this : ↑U' \ U ⊆ ↑U' \ ↑L'
⊢ μ L' + ⨆ x, μ ↑x ≤ μ.outerMeasure ↑U'

Tactic:
]

Goal state after:
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
hL'U : ↑L' ⊆ U
hL'U' : ↑L' ⊆ ↑U'
this : ↑U' \ U ⊆ ↑U' \ ↑L'
⊢ μ L' + ⨆ x, μ ↑x ≤ μ.outerMeasure ↑U'
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝⁶ : LinearOrder R
inst✝⁵ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁴ : OrderTopology R
inst✝³ : CompactIccSpace R
inst✝² : MeasurableSpace R
inst✝¹ : BorelSpace R
inst✝ : DenselyOrdered R
s : Set R
t : ℕ → Set R
ht : s ⊆ ⋃ i, t i
ε : ℝ≥0
ε0 : 0 < ε
h : ∑' (i : ℕ), f.length (t i) < ∞
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑ε
⊢ ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), f.outer t ≤ ∑' (i : ℕ), f.length (t i) + ∑' (i : ℕ), ↑(ε' i)

Tactic:
]

Goal state after:
R : Type u_1
inst✝⁶ : LinearOrder R
inst✝⁵ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁴ : OrderTopology R
inst✝³ : CompactIccSpace R
inst✝² : MeasurableSpace R
inst✝¹ : BorelSpace R
inst✝ : DenselyOrdered R
s : Set R
t : ℕ → Set R
ht : s ⊆ ⋃ i, t i
ε : ℝ≥0
ε0 : 0 < ε
h : ∑' (i : ℕ), f.length (t i) < ∞
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑ε
⊢ ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), f.outer t ≤ ∑' (i : ℕ), f.length (t i) + ∑' (i : ℕ), ↑(ε' i)
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝⁶ : LinearOrder R
inst✝⁵ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁴ : OrderTopology R
inst✝³ : CompactIccSpace R
inst✝² : MeasurableSpace R
inst✝¹ : BorelSpace R
inst✝ : DenselyOrdered R
s : Set R
t : ℕ → Set R
ht : s ⊆ ⋃ i, t i
ε : ℝ≥0
ε0 : 0 < ε
h : ∑' (i : ℕ), f.length (t i) < ∞
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑ε
⊢ ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), f.outer t ≤ ∑' (a : ℕ), (f.length (t a) + ↑(ε' a))

Tactic:
]

Goal state after:
R : Type u_1
inst✝⁶ : LinearOrder R
inst✝⁵ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁴ : OrderTopology R
inst✝³ : CompactIccSpace R
inst✝² : MeasurableSpace R
inst✝¹ : BorelSpace R
inst✝ : DenselyOrdered R
s : Set R
t : ℕ → Set R
ht : s ⊆ ⋃ i, t i
ε : ℝ≥0
ε0 : 0 < ε
h : ∑' (i : ℕ), f.length (t i) < ∞
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑ε
⊢ ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), f.outer t ≤ ∑' (a : ℕ), (f.length (t a) + ↑(ε' a))
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝¹ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
hL'U : ↑L' ⊆ U
hL'U' : ↑L' ⊆ ↑U'
this✝ : ↑U' \ U ⊆ ↑U' \ ↑L'
this : Nonempty { M // ↑M ⊆ ↑U' \ closure ↑L }
⊢ ⨆ i, μ L' + μ ↑i ≤ μ.outerMeasure ↑U'

Tactic:
]

Goal state after:
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝¹ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
hL'U : ↑L' ⊆ U
hL'U' : ↑L' ⊆ ↑U'
this✝ : ↑U' \ U ⊆ ↑U' \ ↑L'
this : Nonempty { M // ↑M ⊆ ↑U' \ closure ↑L }
⊢ ⨆ i, μ L' + μ ↑i ≤ μ.outerMeasure ↑U'
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝¹ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
hL'U : ↑L' ⊆ U
hL'U' : ↑L' ⊆ ↑U'
this✝ : ↑U' \ U ⊆ ↑U' \ ↑L'
this : Nonempty { M // ↑M ⊆ ↑U' \ closure ↑L }
M : Compacts G
hM : ↑M ⊆ ↑U' \ ↑L'
M' : Compacts G := { carrier := closure ↑M, isCompact' := ⋯ }
⊢ μ L' + μ M' ≤ μ.outerMeasure ↑U'

Tactic:
]

Goal state after:
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝¹ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
hL'U : ↑L' ⊆ U
hL'U' : ↑L' ⊆ ↑U'
this✝ : ↑U' \ U ⊆ ↑U' \ ↑L'
this : Nonempty { M // ↑M ⊆ ↑U' \ closure ↑L }
M : Compacts G
hM : ↑M ⊆ ↑U' \ ↑L'
M' : Compacts G := { carrier := closure ↑M, isCompact' := ⋯ }
⊢ μ L' + μ M' ≤ μ.outerMeasure ↑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
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
this : ∀ x ∈ s ∩ k, f x = 0
⊢ ∫ (x : X) in s \ k, f x ∂μ = ∫ (x : X) in s \ k, f x ∂μ

Tactic:
]

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
t : Set α
ht : ∀ (g : G), g • t = t
hts : μ (t ∩ s) = 0
⊢ ∑' (g : G), μ (g • t ∩ s) = 0

Tactic:
]

Goal state after:
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
t : Set α
ht : ∀ (g : G), g • t = t
hts : μ (t ∩ s) = 0
⊢ ∑' (g : G), μ (g • t ∩ s) = 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
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 s, f x ∂μ = ∫ (x : X) in s, f x ∂μ

Tactic:
]

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
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝² : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
hL'U : ↑L' ⊆ U
hL'U' : ↑L' ⊆ ↑U'
this✝¹ : ↑U' \ U ⊆ ↑U' \ ↑L'
this✝ : Nonempty { M // ↑M ⊆ ↑U' \ closure ↑L }
M : Compacts G
hM : ↑M ⊆ ↑U' \ ↑L'
M' : Compacts G := { carrier := closure ↑M, isCompact' := ⋯ }
hM' : ↑M' ⊆ ↑U' \ ↑L'
this : ↑(L' ⊔ M') ⊆ ↑U'
⊢ μ L' + μ M' ≤ μ.innerContent { carrier := ↑U', is_open' := ⋯ }

Tactic:
]

Goal state after:
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝² : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
hL'U : ↑L' ⊆ U
hL'U' : ↑L' ⊆ ↑U'
this✝¹ : ↑U' \ U ⊆ ↑U' \ ↑L'
this✝ : Nonempty { M // ↑M ⊆ ↑U' \ closure ↑L }
M : Compacts G
hM : ↑M ⊆ ↑U' \ ↑L'
M' : Compacts G := { carrier := closure ↑M, isCompact' := ⋯ }
hM' : ↑M' ⊆ ↑U' \ ↑L'
this : ↑(L' ⊔ M') ⊆ ↑U'
⊢ μ L' + μ M' ≤ μ.innerContent { carrier := ↑U', is_open' := ⋯ }
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
α : Type u_3
inst✝⁶ : Group G
inst✝⁵ : MulAction G α
inst✝⁴ : MeasurableSpace α
s : Set α
μ : Measure α
inst✝³ : MeasurableConstSMul G α
inst✝² : SMulInvariantMeasure G α μ
inst✝¹ : Countable G
inst✝ : Finite G
h : IsFundamentalDomain G s μ
t : Set α
ht : ∀ (g : G), g • t =ᵐ[μ] t
this : Fintype G
⊢ ∑' (g : G), μ (g • t ∩ s) = Nat.card G • μ (t ∩ s)

Tactic:
]

Goal state after:
G : Type u_1
α : Type u_3
inst✝⁶ : Group G
inst✝⁵ : MulAction G α
inst✝⁴ : MeasurableSpace α
s : Set α
μ : Measure α
inst✝³ : MeasurableConstSMul G α
inst✝² : SMulInvariantMeasure G α μ
inst✝¹ : Countable G
inst✝ : Finite G
h : IsFundamentalDomain G s μ
t : Set α
ht : ∀ (g : G), g • t =ᵐ[μ] t
this : Fintype G
⊢ ∑' (g : G), μ (g • t ∩ s) = Nat.card G • μ (t ∩ s)
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
A : Set G
hA : MeasurableSet A
r : ℝ≥0∞
hr : ⨅ U, ⨅ (hU : IsOpen U), ⨅ (_ : A ⊆ U), μ.innerContent { carrier := U, is_open' := hU } < r
⊢ ∃ U ⊇ A, IsOpen U ∧ μ.measure U < r

Tactic:
]

Goal state after:
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
A : Set G
hA : MeasurableSet A
r : ℝ≥0∞
hr : ⨅ U, ⨅ (hU : IsOpen U), ⨅ (_ : A ⊆ U), μ.innerContent { carrier := U, is_open' := hU } < r
⊢ ∃ U ⊇ A, IsOpen U ∧ μ.measure U < r
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
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
s t : Set X
μ : Measure X
ht : NullMeasurableSet t μ
hts : s ⊆ t
h't : ∀ᵐ (x : X) ∂μ, x ∈ t \ s → f x = 0
h : ¬IntegrableOn f t μ
this : ¬IntegrableOn f s μ
⊢ 0 = 0

Tactic:
]

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
S : MeasurableSpace G
inst✝ : BorelSpace G
A : Set G
hA : MeasurableSet A
r : ℝ≥0∞
U : Set G
hUo : IsOpen U
hAU : A ⊆ U
hr : μ.measure U < r
⊢ ∃ U ⊇ A, IsOpen U ∧ μ.measure U < r

Tactic:
]

Goal state after:
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
A : Set G
hA : MeasurableSet A
r : ℝ≥0∞
U : Set G
hUo : IsOpen U
hAU : A ⊆ U
hr : μ.measure U < r
⊢ ∃ U ⊇ A, IsOpen U ∧ μ.measure U < r
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : R1Space G
S : MeasurableSpace G
inst✝¹ : BorelSpace G
inst✝ : WeaklyLocallyCompactSpace G
K : Set G
hK : IsCompact K
⊢ μ.outerMeasure (closure K) < ∞

Tactic:
]

Goal state after:
G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : R1Space G
S : MeasurableSpace G
inst✝¹ : BorelSpace G
inst✝ : WeaklyLocallyCompactSpace G
K : Set G
hK : IsCompact K
⊢ μ.outerMeasure (closure K) < ∞
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: case e'_3.e'_4
G : Type u_1
α : Type u_3
inst✝⁵ : Group G
inst✝⁴ : MulAction G α
inst✝³ : MeasurableSpace α
s t : Set α
μ : Measure α
inst✝² : MeasurableConstSMul G α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : Countable G
hs : IsFundamentalDomain G s μ
ht : IsFundamentalDomain G t μ
A : Set α
hA₀ : MeasurableSet A
hA : ∀ (g : G), (fun x ↦ g • x) ⁻¹' A = A
g : G
x : α
⊢ A = A

Tactic:
]

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 : NullMeasurableSet t μ
hts : s ⊆ t
h't : ∀ᵐ (x : X) ∂μ, x ∈ t \ s → f x = 0
h : IntegrableOn f t μ
f' : X → E := AEStronglyMeasurable.mk f ⋯
x : X
hx : x ∈ t \ s → f x = 0
h'x : x ∈ t → f x = AEStronglyMeasurable.mk f ⋯ x
h''x : x ∈ t \ s
⊢ 0 = 0

Tactic:
]

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
S : MeasurableSpace G
inst✝¹ : BorelSpace G
inst✝ : WeaklyLocallyCompactSpace G
this : IsFiniteMeasureOnCompacts μ.measure
U : Set G
hU : IsOpen U
r : ℝ≥0∞
hr : r < μ.innerContent { carrier := U, is_open' := hU }
⊢ ∃ K ⊆ U, IsCompact K ∧ r < μ.measure K

Tactic:
]

Goal state after:
G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : R1Space G
S : MeasurableSpace G
inst✝¹ : BorelSpace G
inst✝ : WeaklyLocallyCompactSpace G
this : IsFiniteMeasureOnCompacts μ.measure
U : Set G
hU : IsOpen U
r : ℝ≥0∞
hr : r < μ.innerContent { carrier := U, is_open' := hU }
⊢ ∃ K ⊆ U, IsCompact K ∧ r < μ.measure K
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inr
R : Type u_1
inst✝⁶ : LinearOrder R
inst✝⁵ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁴ : OrderTopology R
inst✝³ : CompactIccSpace R
inst✝² : MeasurableSpace R
inst✝¹ : BorelSpace R
inst✝ : DenselyOrdered R
s : Set R
t : ℕ → Set R
ht : s ⊆ ⋃ i, t i
ε : ℝ≥0
ε0 : 0 < ε
h : ∑' (i : ℕ), f.length (t i) < ∞
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑ε
i : ℕ
hR : Nonempty R
a b : R
h₁ : t i \ botSet ⊆ Ioc a b
h₂ : f.outer (Ioc a b) < f.length (t i) + ↑(ε' i)
⊢ ∃ s, t i ⊆ s ∧ MeasurableSet s ∧ f.outer s ≤ f.length (t i) + ofReal ↑(ε' i)

Tactic:
]

Goal state after:
case inr
R : Type u_1
inst✝⁶ : LinearOrder R
inst✝⁵ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁴ : OrderTopology R
inst✝³ : CompactIccSpace R
inst✝² : MeasurableSpace R
inst✝¹ : BorelSpace R
inst✝ : DenselyOrdered R
s : Set R
t : ℕ → Set R
ht : s ⊆ ⋃ i, t i
ε : ℝ≥0
ε0 : 0 < ε
h : ∑' (i : ℕ), f.length (t i) < ∞
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑ε
i : ℕ
hR : Nonempty R
a b : R
h₁ : t i \ botSet ⊆ Ioc a b
h₂ : f.outer (Ioc a b) < f.length (t i) + ↑(ε' i)
⊢ ∃ s, t i ⊆ s ∧ MeasurableSet s ∧ f.outer s ≤ f.length (t i) + ofReal ↑(ε' i)
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
α : Type u_3
inst✝⁷ : Group G
inst✝⁶ : MulAction G α
inst✝⁵ : MeasurableSpace α
s t : Set α
μ : Measure α
inst✝⁴ : MeasurableConstSMul G α
inst✝³ : SMulInvariantMeasure G α μ
inst✝² : Countable G
β : Type u_6
inst✝¹ : TopologicalSpace β
inst✝ : PseudoMetrizableSpace β
hs : IsFundamentalDomain G s μ
ht : IsFundamentalDomain G t μ
f : α → β
hf : ∀ (g : G) (x : α), f (g • x) = f x
g : G
he : MeasurableEmbedding fun x ↦ g⁻¹ • x
⊢ AEStronglyMeasurable (f ∘ fun x ↦ g⁻¹ • x) (μ.restrict (g • s ∩ t)) ↔ AEStronglyMeasurable f (μ.restrict (g • s ∩ t))

Tactic:
]

Goal state after:
G : Type u_1
α : Type u_3
inst✝⁷ : Group G
inst✝⁶ : MulAction G α
inst✝⁵ : MeasurableSpace α
s t : Set α
μ : Measure α
inst✝⁴ : MeasurableConstSMul G α
inst✝³ : SMulInvariantMeasure G α μ
inst✝² : Countable G
β : Type u_6
inst✝¹ : TopologicalSpace β
inst✝ : PseudoMetrizableSpace β
hs : IsFundamentalDomain G s μ
ht : IsFundamentalDomain G t μ
f : α → β
hf : ∀ (g : G) (x : α), f (g • x) = f x
g : G
he : MeasurableEmbedding fun x ↦ g⁻¹ • x
⊢ AEStronglyMeasurable (f ∘ fun x ↦ g⁻¹ • x) (μ.restrict (g • s ∩ t)) ↔ AEStronglyMeasurable f (μ.restrict (g • s ∩ t))