mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: γ : Type u_3
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : PolishSpace γ
β : Type u_4
inst✝³ : TopologicalSpace β
inst✝² : T2Space β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
s : Set γ
hs : IsClosed[inst✝⁵] s
f : γ → β
f_cont : ContinuousOn f s
f_inj : InjOn f s
⊢ MeasurableSet (range fun x ↦ f ↑x)

Tactic:
]

Goal state after:
γ : Type u_3
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : PolishSpace γ
β : Type u_4
inst✝³ : TopologicalSpace β
inst✝² : T2Space β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
s : Set γ
hs : IsClosed[inst✝⁵] s
f : γ → β
f_cont : ContinuousOn f s
f_inj : InjOn f s
⊢ MeasurableSet (range fun x ↦ f ↑x)
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: case f_cont
γ : Type u_3
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : PolishSpace γ
β : Type u_4
inst✝³ : TopologicalSpace β
inst✝² : T2Space β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
s : Set γ
hs : IsClosed[inst✝⁵] s
f : γ → β
f_cont : Continuous[instTopologicalSpaceSubtype, inst✝³] (s.domRestrict f)
f_inj : InjOn f s
this : PolishSpace ↑s
⊢ Continuous[instTopologicalSpaceSubtype, inst✝³] fun x ↦ f ↑x

Tactic:
]

Goal state after:
case f_cont
γ : Type u_3
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : PolishSpace γ
β : Type u_4
inst✝³ : TopologicalSpace β
inst✝² : T2Space β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
s : Set γ
hs : IsClosed[inst✝⁵] s
f : γ → β
f_cont : Continuous[instTopologicalSpaceSubtype, inst✝³] (s.domRestrict f)
f_inj : InjOn f s
this : PolishSpace ↑s
⊢ Continuous[instTopologicalSpaceSubtype, inst✝³] fun x ↦ f ↑x
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: case f_inj
γ : Type u_3
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : PolishSpace γ
β : Type u_4
inst✝³ : TopologicalSpace β
inst✝² : T2Space β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
s : Set γ
hs : IsClosed[inst✝⁵] s
f : γ → β
f_cont : ContinuousOn f s
f_inj : Injective (s.domRestrict f)
this : PolishSpace ↑s
⊢ Injective fun x ↦ f ↑x

Tactic:
]

Goal state after:
case f_inj
γ : Type u_3
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : PolishSpace γ
β : Type u_4
inst✝³ : TopologicalSpace β
inst✝² : T2Space β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
s : Set γ
hs : IsClosed[inst✝⁵] s
f : γ → β
f_cont : ContinuousOn f s
f_inj : Injective (s.domRestrict f)
this : PolishSpace ↑s
⊢ Injective fun x ↦ f ↑x
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
ν : Measure β
⊢ (bind 0 fun x ↦ map (Prod.mk x) ν) = 0

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
ν : Measure β
⊢ (bind 0 fun x ↦ map (Prod.mk x) ν) = 0
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: γ : Type u_3
α : Type u_4
inst✝³ : MeasurableSpace α
s : Set γ
f : γ → α
inst✝² : MeasurableSpace.CountablySeparated α
inst✝¹ : MeasurableSpace γ
inst✝ : StandardBorelSpace γ
hs : MeasurableSet s
f_meas : Measurable f
f_inj : InjOn f s
this : UpgradedStandardBorel γ := upgradeStandardBorel γ
tγ : TopologicalSpace γ := inferInstance
τ : TopologicalSpace α
left✝¹ : SecondCountableTopology α
left✝ : T4Space α
right✝ : OpensMeasurableSpace α
t' : TopologicalSpace γ
t't : t' ≤ tγ
f_cont : Continuous[t', τ] f
t'_polish : PolishSpace γ
⊢ MeasurableSet s

Tactic:
]

Goal state after:
γ : Type u_3
α : Type u_4
inst✝³ : MeasurableSpace α
s : Set γ
f : γ → α
inst✝² : MeasurableSpace.CountablySeparated α
inst✝¹ : MeasurableSpace γ
inst✝ : StandardBorelSpace γ
hs : MeasurableSet s
f_meas : Measurable f
f_inj : InjOn f s
this : UpgradedStandardBorel γ := upgradeStandardBorel γ
tγ : TopologicalSpace γ := inferInstance
τ : TopologicalSpace α
left✝¹ : SecondCountableTopology α
left✝ : T4Space α
right✝ : OpensMeasurableSpace α
t' : TopologicalSpace γ
t't : t' ≤ tγ
f_cont : Continuous[t', τ] f
t'_polish : PolishSpace γ
⊢ MeasurableSet s
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
δ : Type u_4
inst✝² : MeasurableSpace δ
f : α → β
g : γ → δ
μa : Measure α
μc : Measure γ
inst✝¹ : SFinite μa
inst✝ : SFinite μc
hf : Measurable f
hg : Measurable g
i : ℕ × ℕ
s : Set β
t : Set δ
hs : MeasurableSet s
ht : MeasurableSet t
⊢ ((sfiniteSeq μa i.1).prod (sfiniteSeq μc i.2)) (Prod.map f g ⁻¹' s ×ˢ t) =
    (sfiniteSeq μa i.1) (f ⁻¹' s) * (sfiniteSeq μc i.2) (g ⁻¹' t)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
δ : Type u_4
inst✝² : MeasurableSpace δ
f : α → β
g : γ → δ
μa : Measure α
μc : Measure γ
inst✝¹ : SFinite μa
inst✝ : SFinite μc
hf : Measurable f
hg : Measurable g
i : ℕ × ℕ
s : Set β
t : Set δ
hs : MeasurableSet s
ht : MeasurableSet t
⊢ ((sfiniteSeq μa i.1).prod (sfiniteSeq μc i.2)) (Prod.map f g ⁻¹' s ×ˢ t) =
    (sfiniteSeq μa i.1) (f ⁻¹' s) * (sfiniteSeq μc i.2) (g ⁻¹' t)
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: α : Type u_1
inst✝³ : MeasurableSpace α
inst✝² : TopologicalSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
H✝ : μ.InnerRegularWRT IsClosed[inst✝²] IsOpen[inst✝²]
hfin : ∀ {s : Set α}, μ s ≠ ∞
s : ℕ → Set α
hsd : Pairwise (Function.onFun Disjoint s)
hsm : ∀ (i : ℕ), MeasurableSet (s i)
H :
  ∀ (i : ℕ) (ε : ℝ≥0∞),
    ε ≠ 0 → ∃ F ⊆ s i, ∃ U ⊇ s i, IsClosed[inst✝²] F ∧ IsOpen[inst✝²] U ∧ μ (s i) ≤ μ F + ε ∧ μ U ≤ μ (s i) + ε
ε : ℝ≥0∞
ε0 : ε ≠ 0
ε0' : ε / 2 ≠ 0
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : ∑' (i : ℕ), δ i < ε / 2
F : ℕ → Set α
hFs : ∀ (n : ℕ), F n ⊆ s n
U : ℕ → Set α
hsU : ∀ (n : ℕ), U n ⊇ s n
hFc : ∀ (n : ℕ), IsClosed[inst✝²] (F n)
hUo : ∀ (n : ℕ), IsOpen[inst✝²] (U n)
hF : ∀ (n : ℕ), μ (s n) ≤ μ (F n) + δ n
hU : ∀ (n : ℕ), μ (U n) ≤ μ (s n) + δ n
⊢ Tendsto (fun t ↦ ∑ k ∈ t, μ (s k) + ε / 2) atTop (𝓝 (∑' (i : ℕ), μ (s i) + ε / 2))

Tactic:
]

Goal state after:
α : Type u_1
inst✝³ : MeasurableSpace α
inst✝² : TopologicalSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
H✝ : μ.InnerRegularWRT IsClosed[inst✝²] IsOpen[inst✝²]
hfin : ∀ {s : Set α}, μ s ≠ ∞
s : ℕ → Set α
hsd : Pairwise (Function.onFun Disjoint s)
hsm : ∀ (i : ℕ), MeasurableSet (s i)
H :
  ∀ (i : ℕ) (ε : ℝ≥0∞),
    ε ≠ 0 → ∃ F ⊆ s i, ∃ U ⊇ s i, IsClosed[inst✝²] F ∧ IsOpen[inst✝²] U ∧ μ (s i) ≤ μ F + ε ∧ μ U ≤ μ (s i) + ε
ε : ℝ≥0∞
ε0 : ε ≠ 0
ε0' : ε / 2 ≠ 0
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : ∑' (i : ℕ), δ i < ε / 2
F : ℕ → Set α
hFs : ∀ (n : ℕ), F n ⊆ s n
U : ℕ → Set α
hsU : ∀ (n : ℕ), U n ⊇ s n
hFc : ∀ (n : ℕ), IsClosed[inst✝²] (F n)
hUo : ∀ (n : ℕ), IsOpen[inst✝²] (U n)
hF : ∀ (n : ℕ), μ (s n) ≤ μ (F n) + δ n
hU : ∀ (n : ℕ), μ (U n) ≤ μ (s n) + δ n
⊢ Tendsto (fun t ↦ ∑ k ∈ t, μ (s k) + ε / 2) atTop (𝓝 (∑' (i : ℕ), μ (s i) + ε / 2))
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: γ : Type u_3
β : Type u_5
inst✝⁶ : MeasurableSpace β
tβ : TopologicalSpace β
inst✝⁵ : T2Space β
s : Set γ
f : γ → β
inst✝⁴ : BorelSpace β
inst✝³ : TopologicalSpace γ
inst✝² : PolishSpace γ
inst✝¹ : MeasurableSpace γ
inst✝ : BorelSpace γ
hs : MeasurableSet s
f_cont : ContinuousOn f s
f_inj : InjOn f s
u : Set ↑s
hu : MeasurableSet u
A : MeasurableSet (Subtype.val '' u)
B : MeasurableSet (f ∘ Subtype.val '' u)
⊢ MeasurableSet (s.domRestrict f '' u)

Tactic:
]

Goal state after:
γ : Type u_3
β : Type u_5
inst✝⁶ : MeasurableSpace β
tβ : TopologicalSpace β
inst✝⁵ : T2Space β
s : Set γ
f : γ → β
inst✝⁴ : BorelSpace β
inst✝³ : TopologicalSpace γ
inst✝² : PolishSpace γ
inst✝¹ : MeasurableSpace γ
inst✝ : BorelSpace γ
hs : MeasurableSet s
f_cont : ContinuousOn f s
f_inj : InjOn f s
u : Set ↑s
hu : MeasurableSet u
A : MeasurableSet (Subtype.val '' u)
B : MeasurableSet (f ∘ Subtype.val '' u)
⊢ MeasurableSet (s.domRestrict f '' u)
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
ν : Measure β
inst✝² : SFinite ν
μ : Measure α
R : Type u_4
inst✝¹ : SMul R ℝ≥0∞
inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
c : R
s : Set (α × β)
hs : MeasurableSet s
⊢ ∫⁻ (x : α), ν (Prod.mk x ⁻¹' s) ∂c • μ = c • ∫⁻ (x : α), ν (Prod.mk x ⁻¹' s) ∂μ

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
ν : Measure β
inst✝² : SFinite ν
μ : Measure α
R : Type u_4
inst✝¹ : SMul R ℝ≥0∞
inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
c : R
s : Set (α × β)
hs : MeasurableSet s
⊢ ∫⁻ (x : α), ν (Prod.mk x ⁻¹' s) ∂c • μ = c • ∫⁻ (x : α), ν (Prod.mk x ⁻¹' s) ∂μ
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: γ : Type u_3
s : Set γ
tγ : TopologicalSpace γ
inst✝¹ : PolishSpace γ
inst✝ : BorelSpace γ
this✝ : MeasurableSpace γ := borel γ
t' : TopologicalSpace γ
t't : t' ≤ tγ
t'_polish : PolishSpace γ
left✝ : IsClosed[t'] s
s_open : IsOpen[t'] s
⊢ MeasurableSet s

pp: γ : Type u_3
s : Set γ
tγ : TopologicalSpace γ
inst✝¹ : PolishSpace γ
inst✝ : BorelSpace γ
this✝ : MeasurableSpace γ := borel γ
t' : TopologicalSpace γ
t't : t' ≤ tγ
t'_polish : PolishSpace γ
left✝ : IsClosed[t'] s
s_open : IsOpen[t'] s
⊢ PolishSpace γ

Tactic:
]

Goal state after:
γ : Type u_3
s : Set γ
tγ : TopologicalSpace γ
inst✝¹ : PolishSpace γ
inst✝ : BorelSpace γ
this✝ : MeasurableSpace γ := borel γ
t' : TopologicalSpace γ
t't : t' ≤ tγ
t'_polish : PolishSpace γ
left✝ : IsClosed[t'] s
s_open : IsOpen[t'] s
⊢ MeasurableSet s

γ : Type u_3
s : Set γ
tγ : TopologicalSpace γ
inst✝¹ : PolishSpace γ
inst✝ : BorelSpace γ
this✝ : MeasurableSpace γ := borel γ
t' : TopologicalSpace γ
t't : t' ≤ tγ
t'_polish : PolishSpace γ
left✝ : IsClosed[t'] s
s_open : IsOpen[t'] s
⊢ PolishSpace γ
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
δ : Type u_4
inst✝² : MeasurableSpace δ
μa : Measure α
μb : Measure β
μc : Measure γ
μd : Measure δ
inst✝¹ : SFinite μa
inst✝ : SFinite μc
f : α → β
hf : MeasurePreserving f μa μb
g : α → γ → δ
hgm : Measurable (uncurry g)
hg : ∀ᵐ (a : α) ∂μa, map (g a) μc = μd
this : Measurable fun p ↦ (f p.1, g p.1 p.2)
h✝ : NeZero μa
a : α
ha : map (g a) μc = μd
⊢ SFinite (map (g a) μc)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
δ : Type u_4
inst✝² : MeasurableSpace δ
μa : Measure α
μb : Measure β
μc : Measure γ
μd : Measure δ
inst✝¹ : SFinite μa
inst✝ : SFinite μc
f : α → β
hf : MeasurePreserving f μa μb
g : α → γ → δ
hgm : Measurable (uncurry g)
hg : ∀ᵐ (a : α) ∂μa, map (g a) μc = μd
this : Measurable fun p ↦ (f p.1, g p.1 p.2)
h✝ : NeZero μa
a : α
ha : map (g a) μc = μd
⊢ SFinite (map (g a) μc)
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: case map_eq.inr
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
δ : Type u_4
inst✝² : MeasurableSpace δ
μa : Measure α
μb : Measure β
μc : Measure γ
μd : Measure δ
inst✝¹ : SFinite μa
inst✝ : SFinite μc
f : α → β
hf : MeasurePreserving f μa μb
g : α → γ → δ
hgm : Measurable (uncurry g)
hg : ∀ᵐ (a : α) ∂μa, map (g a) μc = μd
this : Measurable fun p ↦ (f p.1, g p.1 p.2)
h✝ : NeZero μa
sf : SFinite μd
s : Set (β × δ)
hs : MeasurableSet s
⊢ ∫⁻ (x : α), μc (Prod.mk x ⁻¹' (fun p ↦ (f p.1, g p.1 p.2)) ⁻¹' s) ∂μa = ∫⁻ (a : α), μd (Prod.mk (f a) ⁻¹' s) ∂μa

Tactic:
]

Goal state after:
case map_eq.inr
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
δ : Type u_4
inst✝² : MeasurableSpace δ
μa : Measure α
μb : Measure β
μc : Measure γ
μd : Measure δ
inst✝¹ : SFinite μa
inst✝ : SFinite μc
f : α → β
hf : MeasurePreserving f μa μb
g : α → γ → δ
hgm : Measurable (uncurry g)
hg : ∀ᵐ (a : α) ∂μa, map (g a) μc = μd
this : Measurable fun p ↦ (f p.1, g p.1 p.2)
h✝ : NeZero μa
sf : SFinite μd
s : Set (β × δ)
hs : MeasurableSet s
⊢ ∫⁻ (x : α), μc (Prod.mk x ⁻¹' (fun p ↦ (f p.1, g p.1 p.2)) ⁻¹' s) ∂μa = ∫⁻ (a : α), μd (Prod.mk (f a) ⁻¹' s) ∂μa
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: α : Type u_6
β : Type u_7
t : Set α
g : α → β
inst✝¹⁰ : TopologicalSpace α
inst✝⁹ : MeasurableSpace α
inst✝⁸ : BorelSpace α
inst✝⁷ : LinearOrder α
inst✝⁶ : OrderTopology α
inst✝⁵ : PolishSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
inst✝¹ : LinearOrder β
inst✝ : OrderTopology β
ht : MeasurableSet t
hg : MonotoneOn g t
h'g : ContinuousOn g t
u : Set β := {c | ∃ x y, x ∈ t ∧ y ∈ t ∧ x < y ∧ g x = c ∧ g y = c}
hu : u.Countable
t' : Set α := t ∩ g ⁻¹' u
this : t' = ⋃ c ∈ u, t ∩ g ⁻¹' {c}
⊢ MeasurableSet (⋃ c ∈ u, t ∩ g ⁻¹' {c})

Tactic:
]

Goal state after:
α : Type u_6
β : Type u_7
t : Set α
g : α → β
inst✝¹⁰ : TopologicalSpace α
inst✝⁹ : MeasurableSpace α
inst✝⁸ : BorelSpace α
inst✝⁷ : LinearOrder α
inst✝⁶ : OrderTopology α
inst✝⁵ : PolishSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
inst✝¹ : LinearOrder β
inst✝ : OrderTopology β
ht : MeasurableSet t
hg : MonotoneOn g t
h'g : ContinuousOn g t
u : Set β := {c | ∃ x y, x ∈ t ∧ y ∈ t ∧ x < y ∧ g x = c ∧ g y = c}
hu : u.Countable
t' : Set α := t ∩ g ⁻¹' u
this : t' = ⋃ c ∈ u, t ∩ g ⁻¹' {c}
⊢ MeasurableSet (⋃ c ∈ u, t ∩ g ⁻¹' {c})
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: α : Type u_1
inst✝³ : MeasurableSpace α
inst✝² : TopologicalSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
H✝ : μ.InnerRegularWRT IsClosed[inst✝²] IsOpen[inst✝²]
hfin : ∀ {s : Set α}, μ s ≠ ∞
s : ℕ → Set α
hsd : Pairwise (Function.onFun Disjoint s)
hsm : ∀ (i : ℕ), MeasurableSet (s i)
H :
  ∀ (i : ℕ) (ε : ℝ≥0∞),
    ε ≠ 0 → ∃ F ⊆ s i, ∃ U ⊇ s i, IsClosed[inst✝²] F ∧ IsOpen[inst✝²] U ∧ μ (s i) ≤ μ F + ε ∧ μ U ≤ μ (s i) + ε
ε : ℝ≥0∞
ε0 : ε ≠ 0
ε0' : ε / 2 ≠ 0
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : ∑' (i : ℕ), δ i < ε / 2
F : ℕ → Set α
hFs : ∀ (n : ℕ), F n ⊆ s n
U : ℕ → Set α
hsU : ∀ (n : ℕ), U n ⊇ s n
hFc : ∀ (n : ℕ), IsClosed[inst✝²] (F n)
hUo : ∀ (n : ℕ), IsOpen[inst✝²] (U n)
hF : ∀ (n : ℕ), μ (s n) ≤ μ (F n) + δ n
hU : ∀ (n : ℕ), μ (U n) ≤ μ (s n) + δ n
this : Tendsto (fun t ↦ ∑ k ∈ t, μ (s k) + ε / 2) atTop (𝓝 (μ (⋃ n, s n) + ε / 2))
t : Finset ℕ
ht : μ (⋃ n, s n) < ∑ k ∈ t, μ (s k) + ε / 2
⊢ ∑ k ∈ t, μ (s k) + ε / 2 ≤ ∑ x ∈ t, (μ (F x) + δ x) + ε / 2

Tactic:
]

Goal state after:
α : Type u_1
inst✝³ : MeasurableSpace α
inst✝² : TopologicalSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
H✝ : μ.InnerRegularWRT IsClosed[inst✝²] IsOpen[inst✝²]
hfin : ∀ {s : Set α}, μ s ≠ ∞
s : ℕ → Set α
hsd : Pairwise (Function.onFun Disjoint s)
hsm : ∀ (i : ℕ), MeasurableSet (s i)
H :
  ∀ (i : ℕ) (ε : ℝ≥0∞),
    ε ≠ 0 → ∃ F ⊆ s i, ∃ U ⊇ s i, IsClosed[inst✝²] F ∧ IsOpen[inst✝²] U ∧ μ (s i) ≤ μ F + ε ∧ μ U ≤ μ (s i) + ε
ε : ℝ≥0∞
ε0 : ε ≠ 0
ε0' : ε / 2 ≠ 0
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : ∑' (i : ℕ), δ i < ε / 2
F : ℕ → Set α
hFs : ∀ (n : ℕ), F n ⊆ s n
U : ℕ → Set α
hsU : ∀ (n : ℕ), U n ⊇ s n
hFc : ∀ (n : ℕ), IsClosed[inst✝²] (F n)
hUo : ∀ (n : ℕ), IsOpen[inst✝²] (U n)
hF : ∀ (n : ℕ), μ (s n) ≤ μ (F n) + δ n
hU : ∀ (n : ℕ), μ (U n) ≤ μ (s n) + δ n
this : Tendsto (fun t ↦ ∑ k ∈ t, μ (s k) + ε / 2) atTop (𝓝 (μ (⋃ n, s n) + ε / 2))
t : Finset ℕ
ht : μ (⋃ n, s n) < ∑ k ∈ t, μ (s k) + ε / 2
⊢ ∑ k ∈ t, μ (s k) + ε / 2 ≤ ∑ x ∈ t, (μ (F x) + δ x) + ε / 2
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
δ : Type u_4
inst✝² : MeasurableSpace δ
μa : Measure α
μb : Measure β
μc : Measure γ
μd : Measure δ
inst✝¹ : SFinite μa
inst✝ : SFinite μc
f : α → β
hf : MeasurePreserving f μa μb
g : α → γ → δ
hgm : Measurable (uncurry g)
hg : ∀ᵐ (a : α) ∂μa, map (g a) μc = μd
this : Measurable fun p ↦ (f p.1, g p.1 p.2)
h✝ : NeZero μa
sf : SFinite μd
s : Set (β × δ)
hs : MeasurableSet s
a : α
ha : map (g a) μc = μd
⊢ μc ((fun x ↦ (f (a, x).1, g (a, x).1 (a, x).2)) ⁻¹' s) = μc ((fun x ↦ (f a, g a x)) ⁻¹' s)

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: α : Type u_6
β : Type u_7
t : Set α
g : α → β
inst✝¹⁰ : TopologicalSpace α
inst✝⁹ : MeasurableSpace α
inst✝⁸ : BorelSpace α
inst✝⁷ : LinearOrder α
inst✝⁶ : OrderTopology α
inst✝⁵ : PolishSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
inst✝¹ : LinearOrder β
inst✝ : OrderTopology β
ht : MeasurableSet t
hg : MonotoneOn g t
h'g : ContinuousOn g t
u : Set β := {c | ∃ x y, x ∈ t ∧ y ∈ t ∧ x < y ∧ g x = c ∧ g y = c}
hu : u.Countable
t' : Set α := t ∩ g ⁻¹' u
ht' : MeasurableSet t'
this : g '' t = g '' (t \ t') ∪ g '' t'
⊢ MeasurableSet (g '' (t \ t') ∪ g '' t')

Tactic:
]

Goal state after:
α : Type u_6
β : Type u_7
t : Set α
g : α → β
inst✝¹⁰ : TopologicalSpace α
inst✝⁹ : MeasurableSpace α
inst✝⁸ : BorelSpace α
inst✝⁷ : LinearOrder α
inst✝⁶ : OrderTopology α
inst✝⁵ : PolishSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
inst✝¹ : LinearOrder β
inst✝ : OrderTopology β
ht : MeasurableSet t
hg : MonotoneOn g t
h'g : ContinuousOn g t
u : Set β := {c | ∃ x y, x ∈ t ∧ y ∈ t ∧ x < y ∧ g x = c ∧ g y = c}
hu : u.Countable
t' : Set α := t ∩ g ⁻¹' u
ht' : MeasurableSet t'
this : g '' t = g '' (t \ t') ∪ g '' t'
⊢ MeasurableSet (g '' (t \ t') ∪ g '' t')
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
inst✝¹ : MeasurableSpace γ
f : α × β → γ
μ : Measure α
ν : Measure β
τ : Measure γ
hf : Measurable f
inst✝ : SFinite ν
h2f : ∀ᵐ (x : α) ∂μ, QuasiMeasurePreserving (fun y ↦ f (x, y)) ν τ
s : Set γ
hs : MeasurableSet s
h2s : τ s = 0
⊢ ∫⁻ (x : α), ν (Prod.mk x ⁻¹' f ⁻¹' s) ∂μ = 0

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
inst✝¹ : MeasurableSpace γ
f : α × β → γ
μ : Measure α
ν : Measure β
τ : Measure γ
hf : Measurable f
inst✝ : SFinite ν
h2f : ∀ᵐ (x : α) ∂μ, QuasiMeasurePreserving (fun y ↦ f (x, y)) ν τ
s : Set γ
hs : MeasurableSet s
h2s : τ s = 0
⊢ ∫⁻ (x : α), ν (Prod.mk x ⁻¹' f ⁻¹' s) ∂μ = 0
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
inst✝¹ : MeasurableSpace γ
f : α × β → γ
μ : Measure α
ν : Measure β
τ : Measure γ
hf : Measurable f
inst✝ : SFinite ν
h2f : ∀ᵐ (x : α) ∂μ, QuasiMeasurePreserving (fun y ↦ f (x, y)) ν τ
s : Set γ
hs : MeasurableSet s
h2s : τ s = 0
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: G : Type u_1
H : Type u_2
inst✝³ : MeasurableSpace G
inst✝² : MeasurableSpace H
inst✝¹ : Mul G
μ : Measure G
inst✝ : μ.IsMulLeftInvariant
c : ℝ≥0∞
g : G
⊢ c • μ = c • μ

Tactic:
]

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

Goal state before:
pp: G : Type u_1
H : Type u_2
inst✝³ : MeasurableSpace G
inst✝² : MeasurableSpace H
inst✝¹ : Mul G
μ : Measure G
inst✝ : μ.IsMulRightInvariant
c : ℝ≥0∞
g : G
⊢ c • μ = c • μ

Tactic:
]

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

Goal state before:
pp: α : Type u_4
β : Type u_5
γ : Type u_6
inst✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
f : α × β → γ
μ : Measure α
ν : Measure β
τ : Measure γ
hf : Measurable f
inst✝¹ : SFinite μ
inst✝ : SFinite ν
h2f : ∀ᵐ (y : β) ∂ν, QuasiMeasurePreserving (fun x ↦ f (x, y)) μ τ
⊢ QuasiMeasurePreserving f (map Prod.swap (ν.prod μ)) τ

Tactic:
]

Goal state after:
α : Type u_4
β : Type u_5
γ : Type u_6
inst✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
f : α × β → γ
μ : Measure α
ν : Measure β
τ : Measure γ
hf : Measurable f
inst✝¹ : SFinite μ
inst✝ : SFinite ν
h2f : ∀ᵐ (y : β) ∂ν, QuasiMeasurePreserving (fun x ↦ f (x, y)) μ τ
⊢ QuasiMeasurePreserving f (map Prod.swap (ν.prod μ)) τ
Module: Mathlib.MeasureTheory.Group.Measure

Goal state before:
pp: G : Type u_1
inst✝² : MeasurableSpace G
inst✝¹ : Mul G
inst✝ : MeasurableMul G
μ : Measure G
g : G
A : Set G
hA : MeasurableSet A
⊢ μ ((fun h ↦ g * h) ⁻¹' A) = μ A ↔ μ ((fun x ↦ g * x) ⁻¹' A) = μ A

Tactic:
]

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

Goal state before:
pp: G : Type u_1
inst✝² : MeasurableSpace G
inst✝¹ : Mul G
inst✝ : MeasurableMul G
μ : Measure G
g : G
A : Set G
hA : MeasurableSet A
⊢ μ ((fun h ↦ h * g) ⁻¹' A) = μ A ↔ μ ((fun x ↦ x * g) ⁻¹' A) = μ A

Tactic:
]

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

Goal state before:
pp: G : Type u_1
H✝ : Type u_2
inst✝⁹ : MeasurableSpace G
inst✝⁸ : MeasurableSpace H✝
inst✝⁷ : Mul G
μ : Measure G
inst✝⁶ : MeasurableMul G
inst✝⁵ : μ.IsMulLeftInvariant
inst✝⁴ : SFinite μ
H : Type u_3
inst✝³ : Mul H
mH : MeasurableSpace H
ν : Measure H
inst✝² : MeasurableMul H
inst✝¹ : ν.IsMulLeftInvariant
inst✝ : SFinite ν
g : G
h : H
⊢ μ.prod ν = μ.prod ν

Tactic:
]

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

Goal state before:
pp: G : Type u_1
H✝ : Type u_2
inst✝⁹ : MeasurableSpace G
inst✝⁸ : MeasurableSpace H✝
inst✝⁷ : Mul G
μ : Measure G
inst✝⁶ : MeasurableMul G
inst✝⁵ : μ.IsMulRightInvariant
inst✝⁴ : SFinite μ
H : Type u_3
inst✝³ : Mul H
mH : MeasurableSpace H
ν : Measure H
inst✝² : MeasurableMul H
inst✝¹ : ν.IsMulRightInvariant
inst✝ : SFinite ν
g : G
h : H
⊢ μ.prod ν = μ.prod ν

Tactic:
]

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

Goal state before:
pp: G : Type u_1
inst✝⁶ : MeasurableSpace G
inst✝⁵ : Mul G
μ : Measure G
inst✝⁴ : MeasurableMul G
H : Type u_3
inst✝³ : MeasurableSpace H
inst✝² : Mul H
inst✝¹ : MeasurableMul H
inst✝ : μ.IsMulLeftInvariant
f : G →ₙ* H
hf : Measurable ⇑f
h_surj : Surjective ⇑f
h : H
⊢ Measure.map ((fun x ↦ h * x) ∘ ⇑f) μ = Measure.map (⇑f) μ

Tactic:
]

Goal state after:
G : Type u_1
inst✝⁶ : MeasurableSpace G
inst✝⁵ : Mul G
μ : Measure G
inst✝⁴ : MeasurableMul G
H : Type u_3
inst✝³ : MeasurableSpace H
inst✝² : Mul H
inst✝¹ : MeasurableMul H
inst✝ : μ.IsMulLeftInvariant
f : G →ₙ* H
hf : Measurable ⇑f
h_surj : Surjective ⇑f
h : H
⊢ Measure.map ((fun x ↦ h * x) ∘ ⇑f) μ = Measure.map (⇑f) μ
Module: Mathlib.MeasureTheory.Group.Measure

Goal state before:
pp: G : Type u_1
inst✝⁶ : MeasurableSpace G
inst✝⁵ : Mul G
μ : Measure G
inst✝⁴ : MeasurableMul G
H : Type u_3
inst✝³ : MeasurableSpace H
inst✝² : Mul H
inst✝¹ : MeasurableMul H
inst✝ : μ.IsMulLeftInvariant
f : G →ₙ* H
hf : Measurable ⇑f
h_surj : Surjective ⇑f
g : G
⊢ Measure.map ((fun x ↦ f g * x) ∘ ⇑f) μ = Measure.map (⇑f ∘ fun x ↦ g * x) μ

Tactic:
]

Goal state after:
G : Type u_1
inst✝⁶ : MeasurableSpace G
inst✝⁵ : Mul G
μ : Measure G
inst✝⁴ : MeasurableMul G
H : Type u_3
inst✝³ : MeasurableSpace H
inst✝² : Mul H
inst✝¹ : MeasurableMul H
inst✝ : μ.IsMulLeftInvariant
f : G →ₙ* H
hf : Measurable ⇑f
h_surj : Surjective ⇑f
g : G
⊢ Measure.map ((fun x ↦ f g * x) ∘ ⇑f) μ = Measure.map (⇑f ∘ fun x ↦ g * x) μ
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: α : Type u_6
β : Type u_7
t : Set α
g : α → β
inst✝¹¹ : TopologicalSpace α
inst✝¹⁰ : MeasurableSpace α
inst✝⁹ : BorelSpace α
inst✝⁸ : LinearOrder α
inst✝⁷ : OrderTopology α
inst✝⁶ : PolishSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : MeasurableSpace β
inst✝³ : BorelSpace β
inst✝² : LinearOrder β
inst✝¹ : OrderTopology β
inst✝ : SecondCountableTopology β
ht : MeasurableSet t
hg : MonotoneOn g t
t' : Set α := {x | x ∈ t ∧ ¬ContinuousWithinAt g t x}
ht' : t'.Countable
⊢ g '' t = g '' (t \ t' ∪ t')

Tactic:
]

Goal state after:
α : Type u_6
β : Type u_7
t : Set α
g : α → β
inst✝¹¹ : TopologicalSpace α
inst✝¹⁰ : MeasurableSpace α
inst✝⁹ : BorelSpace α
inst✝⁸ : LinearOrder α
inst✝⁷ : OrderTopology α
inst✝⁶ : PolishSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : MeasurableSpace β
inst✝³ : BorelSpace β
inst✝² : LinearOrder β
inst✝¹ : OrderTopology β
inst✝ : SecondCountableTopology β
ht : MeasurableSet t
hg : MonotoneOn g t
t' : Set α := {x | x ∈ t ∧ ¬ContinuousWithinAt g t x}
ht' : t'.Countable
⊢ g '' t = g '' (t \ t' ∪ t')