mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
⊢ ∫⁻ (a : α) in s, (r • f) a ∂μ = ∫⁻ (a : α) in s, r * f a ∂μ

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
⊢ ∫⁻ (a : α) in s, (r • f) a ∂μ = ∫⁻ (a : α) in s, r * f a ∂μ
Module: Mathlib.MeasureTheory.Group.Prod

Goal state before:
pp: G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
μ ν : Measure G
inst✝⁴ : SFinite ν
inst✝³ : SFinite μ
s : Set G
inst✝² : MeasurableInv G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : ν.IsMulLeftInvariant
sm : MeasurableSet s
f : G → ℝ≥0∞
hf : Measurable f
⊢ (∫⁻ (x : G) in s, 1 ∂μ) * ∫⁻ (y : G), f y ∂ν = ∫⁻ (x : G), ν ((fun z ↦ z * x) ⁻¹' s) * f x⁻¹ ∂μ

Tactic:
← lintegral_indicator sm,

Goal state after:
G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
μ ν : Measure G
inst✝⁴ : SFinite ν
inst✝³ : SFinite μ
s : Set G
inst✝² : MeasurableInv G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : ν.IsMulLeftInvariant
sm : MeasurableSet s
f : G → ℝ≥0∞
hf : Measurable f
⊢ (∫⁻ (a : G), s.indicator (fun x ↦ 1) a ∂μ) * ∫⁻ (y : G), f y ∂ν = ∫⁻ (x : G), ν ((fun z ↦ z * x) ⁻¹' s) * f x⁻¹ ∂μ
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hr : r ≠ ∞
s : Set α
hs : MeasurableSet s
⊢ ∫⁻ (a : α) in s, (r • f) a ∂μ = ∫⁻ (a : α) in s, r * f a ∂μ

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hr : r ≠ ∞
s : Set α
hs : MeasurableSet s
⊢ ∫⁻ (a : α) in s, (r • f) a ∂μ = ∫⁻ (a : α) in s, r * f a ∂μ
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: α : Type u_1
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : TopologicalSpace α
inst✝² : μ.InnerRegularCompactLTTop
inst✝¹ : R1Space α
inst✝ : BorelSpace α
A : Set α
hA : MeasurableSet A
h'A : μ A ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
K : Set α
hKA : K ⊆ A
hK : IsCompact K
hμK : μ A < μ K + ε
⊢ μ A < μ K + ε

Tactic:
]

Goal state after:
α : Type u_1
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : TopologicalSpace α
inst✝² : μ.InnerRegularCompactLTTop
inst✝¹ : R1Space α
inst✝ : BorelSpace α
A : Set α
hA : MeasurableSet A
h'A : μ A ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
K : Set α
hKA : K ⊆ A
hK : IsCompact K
hμK : μ A < μ K + ε
⊢ μ A < μ K + ε
Module: Mathlib.MeasureTheory.Group.Prod

Goal state before:
pp: G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
μ ν : Measure G
inst✝⁴ : SFinite ν
inst✝³ : SFinite μ
s : Set G
inst✝² : MeasurableInv G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : ν.IsMulLeftInvariant
sm : MeasurableSet s
f : G → ℝ≥0∞
hf : Measurable f
⊢ ∫⁻ (x : G), ∫⁻ (y : G), s.indicator (fun x ↦ 1) (y * x) * f x⁻¹ ∂ν ∂μ =
    ∫⁻ (x : G), ν ((fun z ↦ z * x) ⁻¹' s) * f x⁻¹ ∂μ

pp: case hf
G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
μ ν : Measure G
inst✝⁴ : SFinite ν
inst✝³ : SFinite μ
s : Set G
inst✝² : MeasurableInv G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : ν.IsMulLeftInvariant
sm : MeasurableSet s
f : G → ℝ≥0∞
hf : Measurable f
⊢ AEMeasurable (uncurry fun x y ↦ s.indicator (fun x ↦ 1) x * f y) (μ.prod ν)

Tactic:
]

Goal state after:
G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
μ ν : Measure G
inst✝⁴ : SFinite ν
inst✝³ : SFinite μ
s : Set G
inst✝² : MeasurableInv G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : ν.IsMulLeftInvariant
sm : MeasurableSet s
f : G → ℝ≥0∞
hf : Measurable f
⊢ ∫⁻ (x : G), ∫⁻ (y : G), s.indicator (fun x ↦ 1) (y * x) * f x⁻¹ ∂ν ∂μ =
    ∫⁻ (x : G), ν ((fun z ↦ z * x) ⁻¹' s) * f x⁻¹ ∂μ

case hf
G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
μ ν : Measure G
inst✝⁴ : SFinite ν
inst✝³ : SFinite μ
s : Set G
inst✝² : MeasurableInv G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : ν.IsMulLeftInvariant
sm : MeasurableSet s
f : G → ℝ≥0∞
hf : Measurable f
⊢ AEMeasurable (uncurry fun x y ↦ s.indicator (fun x ↦ 1) x * f y) (μ.prod ν)
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : ∫⁻ (a : α), f a ∂μ ≠ ∞
⊢ ∫⁻ (a : α), f a ∂μ ≠ ∞

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : ∫⁻ (a : α), f a ∂μ ≠ ∞
⊢ ∫⁻ (a : α), f a ∂μ ≠ ∞
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
hs₁ : MeasurableSet s
hs₂ : μ s = 0
⊢ ∫⁻ (a : α) in s, f a ∂μ = 0

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
hs₁ : MeasurableSet s
hs₂ : μ s = 0
⊢ ∫⁻ (a : α) in s, f a ∂μ = 0
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
hs : NullMeasurableSet s μ
t : Set α := toMeasurable μ s
A : ∫⁻ (a : α) in t, f a ∂μ = ∫⁻ (a : α) in s, f a ∂μ
B : (μ.withDensity f) t = (μ.withDensity f) s
⊢ (μ.withDensity f) t = ∫⁻ (a : α) in t, f a ∂μ

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
hs : NullMeasurableSet s μ
t : Set α := toMeasurable μ s
A : ∫⁻ (a : α) in t, f a ∂μ = ∫⁻ (a : α) in s, f a ∂μ
B : (μ.withDensity f) t = (μ.withDensity f) s
⊢ (μ.withDensity f) t = ∫⁻ (a : α) in t, f a ∂μ
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ι : Type u_2
inst✝ : Countable ι
f : ι → α → ℝ≥0∞
h : ∀ (i : ι), Measurable (f i)
s : Set α
hs : MeasurableSet s
⊢ ∫⁻ (x : α) in s, (∑' (n : ι), f n) x ∂μ = ∫⁻ (a : α) in s, ∑' (i : ι), f i a ∂μ

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ι : Type u_2
inst✝ : Countable ι
f : ι → α → ℝ≥0∞
h : ∀ (i : ι), Measurable (f i)
s : Set α
hs : MeasurableSet s
⊢ ∫⁻ (x : α) in s, (∑' (n : ι), f n) x ∂μ = ∫⁻ (a : α) in s, ∑' (i : ι), f i a ∂μ
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: α : Type u_1
inst✝⁵ : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace α
inst✝³ : μ.InnerRegularCompactLTTop
inst✝² : IsLocallyFiniteMeasure μ
inst✝¹ : R1Space α
inst✝ : BorelSpace α
K : Set α
hK : IsCompact K
r : ℝ≥0∞
hr : μ K < r
V : Set α
hKV : V ⊇ K
hVo : IsOpen[inst✝⁴] V
hμV : μ V < ∞
this : Fact (μ V < ∞)
U : Set α
hKU : K ⊆ U
hUo : IsOpen[inst✝⁴] U
hμU : μ (U ∩ V) < r
⊢ μ (U ∩ V) < r

Tactic:
]

Goal state after:
α : Type u_1
inst✝⁵ : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace α
inst✝³ : μ.InnerRegularCompactLTTop
inst✝² : IsLocallyFiniteMeasure μ
inst✝¹ : R1Space α
inst✝ : BorelSpace α
K : Set α
hK : IsCompact K
r : ℝ≥0∞
hr : μ K < r
V : Set α
hKV : V ⊇ K
hVo : IsOpen[inst✝⁴] V
hμV : μ V < ∞
this : Fact (μ V < ∞)
U : Set α
hKU : K ⊆ U
hUo : IsOpen[inst✝⁴] U
hμU : μ (U ∩ V) < r
⊢ μ (U ∩ V) < r
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
f : α → ℝ≥0∞
t : Set α
ht : MeasurableSet t
⊢ ((μ.restrict s).withDensity f) t = ((μ.restrict s).withDensity f) t

Tactic:
]

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
⊢ μ.restrict s = μ.restrict s

Tactic:
]

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

Goal state before:
pp: case refine_1
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Measurable f
S : Set α := {x | f x < 0}
hS : MeasurableSet S
⊢ ∀ᵐ (x : α) ∂μ.restrict S, ENNReal.ofReal (f x) = 0 x

Tactic:
]

Goal state after:
case refine_1
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Measurable f
S : Set α := {x | f x < 0}
hS : MeasurableSet S
⊢ ∀ᵐ (x : α) ∂μ.restrict S, ENNReal.ofReal (f x) = 0 x
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: case refine_2
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Measurable f
S : Set α := {x | f x < 0}
hS : MeasurableSet S
⊢ ∀ᵐ (x : α) ∂μ.restrict Sᶜ, ENNReal.ofReal (-f x) = 0 x

Tactic:
]

Goal state after:
case refine_2
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Measurable f
S : Set α := {x | f x < 0}
hS : MeasurableSet S
⊢ ∀ᵐ (x : α) ∂μ.restrict Sᶜ, ENNReal.ofReal (-f x) = 0 x
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
f : α → ℝ≥0∞
t : Set α
ht : MeasurableSet t
⊢ ∫⁻ (a : α) in t ∩ s, f a ∂μ = ∫⁻ (a : α) in t ∩ s, f a ∂μ

Tactic:
]

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

Goal state before:
pp: G : Type u_1
inst✝⁴ : MeasurableSpace G
inst✝³ : TopologicalSpace G
inst✝² : BorelSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
s : Set G
μ : Measure G
t : Set G
kt : s ⊆ t
t_meas : MeasurableSet t
⊢ s * closure[inst✝³] {1} ⊆ t * closure[inst✝³] {1}

Tactic:
]

Goal state after:
G : Type u_1
inst✝⁴ : MeasurableSpace G
inst✝³ : TopologicalSpace G
inst✝² : BorelSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
s : Set G
μ : Measure G
t : Set G
kt : s ⊆ t
t_meas : MeasurableSet t
⊢ s * closure[inst✝³] {1} ⊆ t * closure[inst✝³] {1}
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: α : Type u_1
inst✝⁵ : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace α
inst✝³ : μ.InnerRegularCompactLTTop
inst✝² : IsLocallyFiniteMeasure μ
inst✝¹ : R1Space α
inst✝ : BorelSpace α
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
this : ε / 2 ≠ 0
K : Set α
hKs : K ⊆ s
hKco : IsCompact K
hKcl : IsClosed[inst✝⁴] K
hμK : μ (s \ K) < ε / 2
U : Set α
hKU : K ⊆ U
hUo : IsOpen[inst✝⁴] U
hμU : μ U < μ K + ε / 2
⊢ μ (U \ s) + μ (s \ U) < ε / 2 + ε / 2

Tactic:
]

Goal state after:
α : Type u_1
inst✝⁵ : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace α
inst✝³ : μ.InnerRegularCompactLTTop
inst✝² : IsLocallyFiniteMeasure μ
inst✝¹ : R1Space α
inst✝ : BorelSpace α
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
this : ε / 2 ≠ 0
K : Set α
hKs : K ⊆ s
hKco : IsCompact K
hKcl : IsClosed[inst✝⁴] K
hμK : μ (s \ K) < ε / 2
U : Set α
hKU : K ⊆ U
hUo : IsOpen[inst✝⁴] U
hμU : μ U < μ K + ε / 2
⊢ μ (U \ s) + μ (s \ U) < ε / 2 + ε / 2
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
s : Set α
f : α → ℝ≥0∞
t : Set α
ht : MeasurableSet t
⊢ ∫⁻ (a : α) in t ∩ s, f a ∂μ = ∫⁻ (a : α) in t ∩ s, f a ∂μ

Tactic:
]

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

Goal state before:
pp: α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
⊢ ∫⁻ (a : α) in s, f a ∂μ = ∫⁻ (a : α) in s, f a ∂μ

Tactic:
]

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
⊢ f =ᵐ[μ] 0 ↔ f =ᵐ[μ] 0

Tactic:
]

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
hf : AEMeasurable f μ
hs : (μ.withDensity f) s = 0
t : Set α := toMeasurable (μ.withDensity f) s
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: case mp
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
hf : AEMeasurable f μ
hs : (μ.withDensity f) s = 0
t : Set α := toMeasurable (μ.withDensity f) s
A : μ {a | ¬(a ∈ toMeasurable (μ.withDensity f) s → f a = 0 a)} = 0
⊢ μ ({x | f x ≠ 0} ∩ toMeasurable (μ.withDensity f) s) = 0

pp: case mp
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
hf : AEMeasurable f μ
hs : (μ.withDensity f) s = 0
t : Set α := toMeasurable (μ.withDensity f) s
A : ∀ᵐ (x : α) ∂μ.restrict (toMeasurable (μ.withDensity f) s), f x = 0 x
⊢ NullMeasurableSet (toMeasurable (μ.withDensity f) s) μ

Tactic:
]

Goal state after:
case mp
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
hf : AEMeasurable f μ
hs : (μ.withDensity f) s = 0
t : Set α := toMeasurable (μ.withDensity f) s
A : μ {a | ¬(a ∈ toMeasurable (μ.withDensity f) s → f a = 0 a)} = 0
⊢ μ ({x | f x ≠ 0} ∩ toMeasurable (μ.withDensity f) s) = 0

case mp
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
hf : AEMeasurable f μ
hs : (μ.withDensity f) s = 0
t : Set α := toMeasurable (μ.withDensity f) s
A : ∀ᵐ (x : α) ∂μ.restrict (toMeasurable (μ.withDensity f) s), f x = 0 x
⊢ NullMeasurableSet (toMeasurable (μ.withDensity f) s) μ
Module: Mathlib.MeasureTheory.Group.Measure

Goal state before:
pp: case refine_2
G : Type u_1
inst✝⁴ : MeasurableSpace G
inst✝³ : TopologicalSpace G
inst✝² : BorelSpace G
μ : Measure G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
h : μ.InnerRegularCompactLTTop
s : Set G
s_meas : MeasurableSet s
μs : μ s ≠ ∞
r : ℝ≥0∞
hr : r < μ s
K : Set G
Ks : K ⊆ s
K_comp : IsCompact K
hK : r < μ K
⊢ r < μ K

Tactic:
]

Goal state after:
case refine_2
G : Type u_1
inst✝⁴ : MeasurableSpace G
inst✝³ : TopologicalSpace G
inst✝² : BorelSpace G
μ : Measure G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
h : μ.InnerRegularCompactLTTop
s : Set G
s_meas : MeasurableSet s
μs : μ s ≠ ∞
r : ℝ≥0∞
hr : r < μ s
K : Set G
Ks : K ⊆ s
K_comp : IsCompact K
hK : r < μ K
⊢ r < μ K
Module: Mathlib.MeasureTheory.Group.Measure

Goal state before:
pp: G : Type u_1
inst✝¹⁰ : MeasurableSpace G
inst✝⁹ : Group G
inst✝⁸ : TopologicalSpace G
μ : Measure G
inst✝⁷ : μ.IsHaarMeasure
inst✝⁶ : BorelSpace G
inst✝⁵ : ContinuousMul G
H : Type u_3
inst✝⁴ : Group H
inst✝³ : TopologicalSpace H
inst✝² : MeasurableSpace H
inst✝¹ : BorelSpace H
inst✝ : IsTopologicalGroup H
f : G →* H
hf : Continuous[inst✝⁸, inst✝³] ⇑f
h_surj : Surjective ⇑f
h_prop : Tendsto (⇑f) (cocompact G) (cocompact H)
K : Set H
hK : IsCompact K
⊢ μ (⇑f ⁻¹' closure[inst✝³] K) < ∞

Tactic:
]

Goal state after:
G : Type u_1
inst✝¹⁰ : MeasurableSpace G
inst✝⁹ : Group G
inst✝⁸ : TopologicalSpace G
μ : Measure G
inst✝⁷ : μ.IsHaarMeasure
inst✝⁶ : BorelSpace G
inst✝⁵ : ContinuousMul G
H : Type u_3
inst✝⁴ : Group H
inst✝³ : TopologicalSpace H
inst✝² : MeasurableSpace H
inst✝¹ : BorelSpace H
inst✝ : IsTopologicalGroup H
f : G →* H
hf : Continuous[inst✝⁸, inst✝³] ⇑f
h_surj : Surjective ⇑f
h_prop : Tendsto (⇑f) (cocompact G) (cocompact H)
K : Set H
hK : IsCompact K
⊢ μ (⇑f ⁻¹' closure[inst✝³] K) < ∞
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: α : Type u_1
inst✝⁵ : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace α
inst✝³ : μ.InnerRegularCompactLTTop
inst✝² : IsLocallyFiniteMeasure μ
inst✝¹ : R1Space α
inst✝ : BorelSpace α
s : Set α
hμs : μ s ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
t : Set α
htm : MeasurableSet t
hst : s =ᵐ[μ] t
⊢ μ s ≠ ∞

Tactic:
]

Goal state after:
α : Type u_1
inst✝⁵ : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace α
inst✝³ : μ.InnerRegularCompactLTTop
inst✝² : IsLocallyFiniteMeasure μ
inst✝¹ : R1Space α
inst✝ : BorelSpace α
s : Set α
hμs : μ s ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
t : Set α
htm : MeasurableSet t
hst : s =ᵐ[μ] t
⊢ μ s ≠ ∞
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: α : Type u_1
inst✝⁵ : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace α
inst✝³ : μ.InnerRegularCompactLTTop
inst✝² : IsLocallyFiniteMeasure μ
inst✝¹ : R1Space α
inst✝ : BorelSpace α
s : Set α
hμs : μ s ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
t : Set α
htm : MeasurableSet t
hst : s =ᵐ[μ] t
U : Set α
hUo : IsOpen[inst✝⁴] U
hμU : μ U < ∞
hUs : μ (U ∆ t) < ε
⊢ μ (U ∆ t) < ε

Tactic:
]

Goal state after:
α : Type u_1
inst✝⁵ : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace α
inst✝³ : μ.InnerRegularCompactLTTop
inst✝² : IsLocallyFiniteMeasure μ
inst✝¹ : R1Space α
inst✝ : BorelSpace α
s : Set α
hμs : μ s ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
t : Set α
htm : MeasurableSet t
hst : s =ᵐ[μ] t
U : Set α
hUo : IsOpen[inst✝⁴] U
hμU : μ U < ∞
hUs : μ (U ∆ t) < ε
⊢ μ (U ∆ t) < ε
Module: Mathlib.MeasureTheory.Group.Measure

Goal state before:
pp: G : Type u_1
H : Type u_2
inst✝¹⁰ : MeasurableSpace G
inst✝⁹ : MeasurableSpace H
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
μ : Measure G
inst✝⁶ : μ.IsHaarMeasure
α : Type u_3
inst✝⁵ : BorelSpace G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : Group α
inst✝² : MulAction α G
inst✝¹ : SMulCommClass α G G
inst✝ : ContinuousConstSMul α G
a : α
K : Set G
hK : IsCompact K
F : G ≃ᵐ G := (Homeomorph.smul a).toMeasurableEquiv
⊢ μ (⇑F ⁻¹' K) < ∞

Tactic:
]

Goal state after:
G : Type u_1
H : Type u_2
inst✝¹⁰ : MeasurableSpace G
inst✝⁹ : MeasurableSpace H
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
μ : Measure G
inst✝⁶ : μ.IsHaarMeasure
α : Type u_3
inst✝⁵ : BorelSpace G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : Group α
inst✝² : MulAction α G
inst✝¹ : SMulCommClass α G G
inst✝ : ContinuousConstSMul α G
a : α
K : Set G
hK : IsCompact K
F : G ≃ᵐ G := (Homeomorph.smul a).toMeasurableEquiv
⊢ μ (⇑F ⁻¹' K) < ∞
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
hf : AEMeasurable f μ
hs : μ ({x | f x ≠ 0} ∩ s) = 0
t : Set α := toMeasurable μ ({x | f x ≠ 0} ∩ s)
A : s ⊆ t ∪ {x | f x = 0}
⊢ μ ({x | f x ≠ 0} ∩ s) = 0

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
hf : AEMeasurable f μ
hs : μ ({x | f x ≠ 0} ∩ s) = 0
t : Set α := toMeasurable μ ({x | f x ≠ 0} ∩ s)
A : s ⊆ t ∪ {x | f x = 0}
⊢ μ ({x | f x ≠ 0} ∩ s) = 0
Module: Mathlib.MeasureTheory.Group.Prod

Goal state before:
pp: G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
s✝ : Set G
inst✝⁴ : MeasurableInv G
μ' ν' : Measure G
inst✝³ : SigmaFinite μ'
inst✝² : SigmaFinite ν'
inst✝¹ : μ'.IsMulLeftInvariant
inst✝ : ν'.IsMulLeftInvariant
s : Set G
hμs : μ' s ≠ ∞
sm : MeasurableSet s
A : Set G
hA : MeasurableSet A
a✝ : μ' A < ∞
h3A : ν' A⁻¹ < ∞
h1 : μ' s * ∫⁻ (a : G) in A⁻¹, 1 a ∂ν' = ∫⁻ (x : G), ν' ((fun z ↦ z * x) ⁻¹' s) * A⁻¹.indicator 1 x⁻¹ ∂μ'
⊢ ∫⁻ (x : G) in A, ν' ((fun y ↦ y * x) ⁻¹' s) ∂μ' ≠ ∞

Tactic:
]

Goal state after:
G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
s✝ : Set G
inst✝⁴ : MeasurableInv G
μ' ν' : Measure G
inst✝³ : SigmaFinite μ'
inst✝² : SigmaFinite ν'
inst✝¹ : μ'.IsMulLeftInvariant
inst✝ : ν'.IsMulLeftInvariant
s : Set G
hμs : μ' s ≠ ∞
sm : MeasurableSet s
A : Set G
hA : MeasurableSet A
a✝ : μ' A < ∞
h3A : ν' A⁻¹ < ∞
h1 : μ' s * ∫⁻ (a : G) in A⁻¹, 1 a ∂ν' = ∫⁻ (x : G), ν' ((fun z ↦ z * x) ⁻¹' s) * A⁻¹.indicator 1 x⁻¹ ∂μ'
⊢ ∫⁻ (x : G) in A, ν' ((fun y ↦ y * x) ⁻¹' s) ∂μ' ≠ ∞
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
hs : μ ({x | f x ≠ 0} ∩ s) = 0
t : Set α := toMeasurable μ ({x | f x ≠ 0} ∩ s)
A : s ⊆ t ∪ {x | f x = 0}
g : α → ℝ≥0∞
hg : Measurable g
hfg : f =ᵐ[μ] g
a : α
ha : f a = g a
⊢ {x | f x = 0} a ↔ {x | g x = 0} a

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
hs : μ ({x | f x ≠ 0} ∩ s) = 0
t : Set α := toMeasurable μ ({x | f x ≠ 0} ∩ s)
A : s ⊆ t ∪ {x | f x = 0}
g : α → ℝ≥0∞
hg : Measurable g
hfg : f =ᵐ[μ] g
a : α
ha : f a = g a
⊢ {x | f x = 0} a ↔ {x | g x = 0} a
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
hs : μ ({x | f x ≠ 0} ∩ s) = 0
t : Set α := toMeasurable μ ({x | f x ≠ 0} ∩ s)
A : s ⊆ t ∪ {x | f x = 0}
g : α → ℝ≥0∞
hg : Measurable g
hfg : f =ᵐ[μ] g
a : α
ha : 0 = g a
h : {x | f x = 0} a
⊢ {x | g x = 0} a

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
hs : μ ({x | f x ≠ 0} ∩ s) = 0
t : Set α := toMeasurable μ ({x | f x ≠ 0} ∩ s)
A : s ⊆ t ∪ {x | f x = 0}
g : α → ℝ≥0∞
hg : Measurable g
hfg : f =ᵐ[μ] g
a : α
ha : 0 = g a
h : {x | f x = 0} a
⊢ {x | g x = 0} a
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
hs : μ ({x | f x ≠ 0} ∩ s) = 0
t : Set α := toMeasurable μ ({x | f x ≠ 0} ∩ s)
A : s ⊆ t ∪ {x | f x = 0}
g : α → ℝ≥0∞
hg : Measurable g
hfg : f =ᵐ[μ] g
a : α
ha : f a = 0
h : {x | g x = 0} a
⊢ {x | f x = 0} a

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
hs : μ ({x | f x ≠ 0} ∩ s) = 0
t : Set α := toMeasurable μ ({x | f x ≠ 0} ∩ s)
A : s ⊆ t ∪ {x | f x = 0}
g : α → ℝ≥0∞
hg : Measurable g
hfg : f =ᵐ[μ] g
a : α
ha : f a = 0
h : {x | g x = 0} a
⊢ {x | f x = 0} a
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace α
inst✝³ : BorelSpace α
inst✝² : MeasurableSpace β
inst✝¹ : TopologicalSpace β
inst✝ : BorelSpace β
h : μ.InnerRegularCompactLTTop
f : α → β
hf : Continuous[inst✝⁴, inst✝¹] f
s : Set β
hs : MeasurableSet s
h's : μ (f ⁻¹' s) ≠ ∞
⊢ μ (f ⁻¹' s) ≠ ∞

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace α
inst✝³ : BorelSpace α
inst✝² : MeasurableSpace β
inst✝¹ : TopologicalSpace β
inst✝ : BorelSpace β
h : μ.InnerRegularCompactLTTop
f : α → β
hf : Continuous[inst✝⁴, inst✝¹] f
s : Set β
hs : MeasurableSet s
h's : μ (f ⁻¹' s) ≠ ∞
⊢ μ (f ⁻¹' s) ≠ ∞
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
hs : μ ({x | f x ≠ 0} ∩ s) = 0
t✝ : Set α := toMeasurable μ ({x | f x ≠ 0} ∩ s)
A : s ⊆ t✝ ∪ {x | f x = 0}
g : α → ℝ≥0∞
hg : Measurable g
hfg : f =ᵐ[μ] g
t : {x | f x = 0} =ᵐ[μ.withDensity f] {x | g x = 0}
⊢ (μ.withDensity g) {x | g x = 0} = 0

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
hs : μ ({x | f x ≠ 0} ∩ s) = 0
t✝ : Set α := toMeasurable μ ({x | f x ≠ 0} ∩ s)
A : s ⊆ t✝ ∪ {x | f x = 0}
g : α → ℝ≥0∞
hg : Measurable g
hfg : f =ᵐ[μ] g
t : {x | f x = 0} =ᵐ[μ.withDensity f] {x | g x = 0}
⊢ (μ.withDensity g) {x | g x = 0} = 0
Module: Mathlib.MeasureTheory.Group.Prod

Goal state before:
pp: G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
s✝ : Set G
inst✝⁴ : MeasurableInv G
μ' ν' : Measure G
inst✝³ : SigmaFinite μ'
inst✝² : SigmaFinite ν'
inst✝¹ : μ'.IsMulLeftInvariant
inst✝ : ν'.IsMulLeftInvariant
s : Set G
hμs : μ' s ≠ ∞
sm : MeasurableSet s
A : Set G
hA : MeasurableSet A
a✝ : μ' A < ∞
h3A : ν' A⁻¹ < ∞
h1 : μ' s * ν' A⁻¹ = ∫⁻ (x : G), A.indicator (fun j ↦ ν' ((fun x ↦ x * j) ⁻¹' s)) x ∂μ'
⊢ μ' s * ν' A⁻¹ ≠ ∞

Tactic:
]

Goal state after:
G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
s✝ : Set G
inst✝⁴ : MeasurableInv G
μ' ν' : Measure G
inst✝³ : SigmaFinite μ'
inst✝² : SigmaFinite ν'
inst✝¹ : μ'.IsMulLeftInvariant
inst✝ : ν'.IsMulLeftInvariant
s : Set G
hμs : μ' s ≠ ∞
sm : MeasurableSet s
A : Set G
hA : MeasurableSet A
a✝ : μ' A < ∞
h3A : ν' A⁻¹ < ∞
h1 : μ' s * ν' A⁻¹ = ∫⁻ (x : G), A.indicator (fun j ↦ ν' ((fun x ↦ x * j) ⁻¹' s)) x ∂μ'
⊢ μ' s * ν' A⁻¹ ≠ ∞
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
hs : μ ({x | f x ≠ 0} ∩ s) = 0
t✝ : Set α := toMeasurable μ ({x | f x ≠ 0} ∩ s)
A : s ⊆ t✝ ∪ {x | f x = 0}
g : α → ℝ≥0∞
hg : Measurable g
hfg : f =ᵐ[μ] g
t : {x | f x = 0} =ᵐ[μ.withDensity f] {x | g x = 0}
M : MeasurableSet {x | g x = 0}
⊢ g =ᵐ[μ.restrict {x | g x = 0}] 0

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
hs : μ ({x | f x ≠ 0} ∩ s) = 0
t✝ : Set α := toMeasurable μ ({x | f x ≠ 0} ∩ s)
A : s ⊆ t✝ ∪ {x | f x = 0}
g : α → ℝ≥0∞
hg : Measurable g
hfg : f =ᵐ[μ] g
t : {x | f x = 0} =ᵐ[μ.withDensity f] {x | g x = 0}
M : MeasurableSet {x | g x = 0}
⊢ g =ᵐ[μ.restrict {x | g x = 0}] 0