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

Goal state before:
pp: Ω : Type u_1
inst✝ : MeasurableSpace Ω
μ : FiniteMeasure Ω
μ_mass : μ.mass = 0
⊢ μ = 0

Tactic:
apply toMeasure_injective

Goal state after:
Ω : Type u_1
inst✝ : MeasurableSpace Ω
μ : FiniteMeasure Ω
μ_mass : μ.mass = 0
⊢ ↑μ = ↑0
Module: Mathlib.MeasureTheory.Measure.FiniteMeasure

Goal state before:
pp: Ω : Type u_1
inst✝ : MeasurableSpace Ω
⊢ {μ | IsFiniteMeasure μ} = (fun μ ↦ μ univ) ⁻¹' Ico 0 ∞

Tactic:
ext μ

Goal state after:
Ω : Type u_1
inst✝ : MeasurableSpace Ω
μ : Measure Ω
⊢ μ ∈ {μ | IsFiniteMeasure μ} ↔ μ ∈ (fun μ ↦ μ univ) ⁻¹' Ico 0 ∞
Module: Mathlib.MeasureTheory.Measure.FiniteMeasure

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : TopologicalSpace Ω
μ : FiniteMeasure Ω
f g : Ω →ᵇ ℝ≥0
f_le_g : ⇑f ≤ ⇑g
⊢ μ.testAgainstNN f ≤ μ.testAgainstNN g

Tactic:
by
  simp only [← ENNReal.coe_le_coe, testAgainstNN_coe_eq]
  gcongr
  apply f_le_g

Goal state after:
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ∞
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
C : ℝ≥0
hC : ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal 1
i : ι
x : α
hx : x ∈ {x | ‖f i x‖₊ < C}
⊢ ‖f i x‖ₑ ≤ ‖f i x + 0‖ₑ

pp: case pos.h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ∞
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
C : ℝ≥0
hC : ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal 1
i : ι
x : α
hx : x ∈ {x | ‖f i x‖₊ < C}
⊢ x ∉ {x | C ≤ ‖f i x‖₊}

Tactic:
add_zero

Goal state after:
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ∞
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
C : ℝ≥0
hC : ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal 1
i : ι
x : α
hx : x ∈ {x | ‖f i x‖₊ < C}
⊢ ‖f i x‖ₑ ≤ ‖f i x‖ₑ

case pos.h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ∞
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
C : ℝ≥0
hC : ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal 1
i : ι
x : α
hx : x ∈ {x | ‖f i x‖₊ < C}
⊢ x ∉ {x | C ≤ ‖f i x‖₊}
Module: Mathlib.MeasureTheory.Measure.Tight

Goal state before:
pp: 𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
S : Set (Measure 𝓧)
inst✝ : TopologicalSpace 𝓧
⊢ IsTightMeasureSet S ↔ ∀ (ε : ENNReal), 0 < ε → ∃ K, IsCompact K ∧ ∀ μ ∈ S, μ Kᶜ ≤ ε

Tactic:
simp only [IsTightMeasureSet, ENNReal.tendsto_nhds ENNReal.zero_ne_top, gt_iff_lt, zero_add,
    iSup_apply, mem_Icc, tsub_le_iff_right, zero_le, iSup_le_iff, true_and, eventually_smallSets,
    mem_cocompact]

Goal state after:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
S : Set (Measure 𝓧)
inst✝ : TopologicalSpace 𝓧
⊢ (∀ (ε : ENNReal), 0 < ε → ∃ s, (∃ t, IsCompact t ∧ tᶜ ⊆ s) ∧ ∀ t ⊆ s, ∀ i ∈ S, i t ≤ ε) ↔
    ∀ (ε : ENNReal), 0 < ε → ∃ K, IsCompact K ∧ ∀ μ ∈ S, μ Kᶜ ≤ ε
Module: Mathlib.MeasureTheory.Measure.Portmanteau

Goal state before:
pp: case inr
Ω : Type u_1
inst✝² : MeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
μs : ι → Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
E : Set Ω
E_mble : MeasurableSet E
h : limsup (fun i ↦ (μs i) E) L ≤ μ E
hne : L.NeBot
meas_Ec : μ Eᶜ = 1 - μ E
meas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E
key : 1 - limsup (fun i ↦ (μs i) E) L = liminf ((fun x ↦ 1 - x) ∘ fun i ↦ (μs i) E) L
⊢ 1 - μ E ≤ liminf ((fun x ↦ 1 - x) ∘ fun i ↦ (μs i) E) L

Tactic:
simpa [← key] using antitone_const_tsub h

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

Goal state before:
pp: case inr
Ω : Type u_1
inst✝² : MeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
μs : ι → Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
E : Set Ω
E_mble : MeasurableSet E
h : μ E ≤ liminf (fun i ↦ (μs i) E) L
hne : L.NeBot
meas_Ec : μ Eᶜ = 1 - μ E
meas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E
key : 1 - liminf (fun i ↦ (μs i) E) L = limsup ((fun x ↦ 1 - x) ∘ fun i ↦ (μs i) E) L
⊢ limsup ((fun x ↦ 1 - x) ∘ fun i ↦ (μs i) E) L ≤ 1 - μ E

Tactic:
simpa [← key] using antitone_const_tsub h

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

Goal state before:
pp: 𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
inst✝¹ : TopologicalSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
inst✝ : TopologicalSpace 𝓨
μ : Set (Measure (𝓧 × 𝓨))
hμ₁ : ∀ (ε : ENNReal), 0 < ε → ∃ K, IsCompact K ∧ ∀ μ_1 ∈ Measure.fst '' μ, μ_1 Kᶜ ≤ ε
hμ₂ : ∀ (ε : ENNReal), 0 < ε → ∃ K, IsCompact K ∧ ∀ μ_1 ∈ Measure.snd '' μ, μ_1 Kᶜ ≤ ε
ε : ENNReal
hε : 0 < ε
K₁ : Set 𝓧
hK₁_compact : IsCompact K₁
K₂ : Set 𝓨
hK₂_compact : IsCompact K₂
hK₂_le : ∀ μ_1 ∈ Measure.snd '' μ, μ_1 K₂ᶜ ≤ ε / 2
κ : Measure (𝓧 × 𝓨)
hK₁_le : (Measure.map Prod.fst κ) K₁ᶜ ≤ ε / 2
hκ_mem : κ ∈ μ
⊢ AEMeasurable Prod.fst κ

Tactic:
by fun_prop

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

Goal state before:
pp: 𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
inst✝¹ : TopologicalSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
inst✝ : TopologicalSpace 𝓨
μ : Set (Measure (𝓧 × 𝓨))
hμ₁ : ∀ (ε : ENNReal), 0 < ε → ∃ K, IsCompact K ∧ ∀ μ_1 ∈ Measure.fst '' μ, μ_1 Kᶜ ≤ ε
hμ₂ : ∀ (ε : ENNReal), 0 < ε → ∃ K, IsCompact K ∧ ∀ μ_1 ∈ Measure.snd '' μ, μ_1 Kᶜ ≤ ε
ε : ENNReal
hε : 0 < ε
K₁ : Set 𝓧
hK₁_compact : IsCompact K₁
hK₁_le : ∀ μ_1 ∈ Measure.fst '' μ, μ_1 K₁ᶜ ≤ ε / 2
K₂ : Set 𝓨
hK₂_compact : IsCompact K₂
κ : Measure (𝓧 × 𝓨)
hK₂_le : (Measure.map Prod.snd κ) K₂ᶜ ≤ ε / 2
hκ_mem : κ ∈ μ
⊢ AEMeasurable Prod.snd κ

Tactic:
by fun_prop

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

Goal state before:
pp: case refine_2
Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : SFinite μ
s : Set Ω
Rs : ℕ → ℝ
left✝ : StrictAnti Rs
Rs_pos : ∀ (n : ℕ), 0 < Rs n
Rs_lim : Tendsto Rs atTop (𝓝 0)
obs : ∀ (n : ℕ), ∃ r ∈ Ioo 0 (Rs n), μ (frontier (Metric.thickening r s)) = 0
⊢ ∀ (n : ℕ), 0 < (fun n ↦ ⋯.choose) n ∧ μ (frontier (Metric.thickening ((fun n ↦ ⋯.choose) n) s)) = 0

Tactic:
exact fun n ↦ ⟨(obs n).choose_spec.1.1, (obs n).choose_spec.2⟩

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

Goal state before:
pp: Ω' : Type u_2
inst✝⁶ : MeasurableSpace Ω'
inst✝⁵ : TopologicalSpace Ω'
inst✝⁴ : BorelSpace Ω'
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : BorelSpace Ω
inst✝ : NormalSpace Ω'
f : Ω → Ω'
hf : IsClosedEmbedding f
M : Set (FiniteMeasure Ω') := {μ | μ (range f)ᶜ = 0}
A : IsEmbedding Subtype.val
⊢ Continuous[instTopologicalSpace, instTopologicalSpaceSubtype] fun μ ↦ ⟨μ.map f, ⋯⟩

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Independence.Kernel.Indep

Goal state before:
pp: α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
f : ι → Set Ω
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
⊢ iIndep (fun i ↦ MeasurableSpace.comap (fun x ↦ x ∈ f i) ⊤) κ μ ↔ iIndepSet f κ μ

Tactic:
simp_rw [← generateFrom_singleton, iIndepSet]

Goal state after:
Module: Mathlib.Probability.Independence.Basic

Goal state before:
pp: Ω : Type u_10
Ω' : Type u_11
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
μ : Measure Ω
ν : Measure Ω'
inst✝³ : IsProbabilityMeasure μ
inst✝² : IsProbabilityMeasure ν
𝓧 : Type u_12
𝓨 : Type u_13
inst✝¹ : MeasurableSpace 𝓧
inst✝ : MeasurableSpace 𝓨
X : Ω → 𝓧
Y : Ω' → 𝓨
mX : Measurable X
mY : Measurable Y
⊢ AEMeasurable (fun ω ↦ X ω.1) (μ.prod ν)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Independence.Basic

Goal state before:
pp: Ω : Type u_10
Ω' : Type u_11
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
μ : Measure Ω
ν : Measure Ω'
inst✝³ : IsProbabilityMeasure μ
inst✝² : IsProbabilityMeasure ν
𝓧 : Type u_12
𝓨 : Type u_13
inst✝¹ : MeasurableSpace 𝓧
inst✝ : MeasurableSpace 𝓨
X : Ω → 𝓧
Y : Ω' → 𝓨
mX : Measurable X
mY : Measurable Y
⊢ AEMeasurable (fun ω ↦ Y ω.2) (μ.prod ν)

Tactic:
by fun_prop

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

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : ProbabilityMeasure Ω
μs : ℕ → ProbabilityMeasure Ω
F : Set Ω
h : limsup (fun i ↦ (μs i) F) atTop ≤ μ F
hF_closed : IsClosed[inst✝¹] F
⊢ ContinuousAt ENNReal.ofNNReal (limsup (fun x ↦ (μs x) F) atTop)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Independence.Kernel.IndepFun

Goal state before:
pp: case refine_1
α : Type u_1
Ω : Type u_2
β : Type u_4
γ : Type u_6
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteKernel κ
f : Ω → β
g : Ω → γ
hf : Measurable f
hg : Measurable g
h :
  ∀ (s : Set β) (t : Set γ),
    MeasurableSet s → MeasurableSet t → ∀ᵐ (a : α) ∂μ, (κ a) (f ⁻¹' s ∩ g ⁻¹' t) = (κ a) (f ⁻¹' s) * (κ a) (g ⁻¹' t)
u : Set α
s : Set β
t : Set γ
hu : MeasurableSet u
hs : MeasurableSet s
ht : MeasurableSet t
⊢ ∫⁻ (a : α), ((κ.map fun ω ↦ (f ω, g ω)) a) (Prod.mk a ⁻¹' u ×ˢ s ×ˢ t) ∂μ =
    ∫⁻ (a : α), ((κ.map f ×ₖ κ.map g) a) (Prod.mk a ⁻¹' u ×ˢ s ×ˢ t) ∂μ

Tactic:
refine lintegral_congr_ae ?_

Goal state after:
case refine_1
α : Type u_1
Ω : Type u_2
β : Type u_4
γ : Type u_6
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteKernel κ
f : Ω → β
g : Ω → γ
hf : Measurable f
hg : Measurable g
h :
  ∀ (s : Set β) (t : Set γ),
    MeasurableSet s → MeasurableSet t → ∀ᵐ (a : α) ∂μ, (κ a) (f ⁻¹' s ∩ g ⁻¹' t) = (κ a) (f ⁻¹' s) * (κ a) (g ⁻¹' t)
u : Set α
s : Set β
t : Set γ
hu : MeasurableSet u
hs : MeasurableSet s
ht : MeasurableSet t
⊢ (fun a ↦ ((κ.map fun ω ↦ (f ω, g ω)) a) (Prod.mk a ⁻¹' u ×ˢ s ×ˢ t)) =ᵐ[μ] fun a ↦
    ((κ.map f ×ₖ κ.map g) a) (Prod.mk a ⁻¹' u ×ˢ s ×ˢ t)
Module: Mathlib.Probability.Independence.Kernel.IndepFun

Goal state before:
pp: α : Type u_1
Ω : Type u_2
β : Type u_4
γ : Type u_6
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteKernel κ
f : Ω → β
g : Ω → γ
hf : Measurable f
hg : Measurable g
h :
  ∀ (s : Set β) (t : Set γ),
    MeasurableSet s → MeasurableSet t → ∀ᵐ (a : α) ∂μ, (κ a) (f ⁻¹' s ∩ g ⁻¹' t) = (κ a) (f ⁻¹' s) * (κ a) (g ⁻¹' t)
u : Set α
s : Set β
t : Set γ
hu : MeasurableSet u
hs : MeasurableSet s
ht : MeasurableSet t
h_set_eq : ∀ (ω : α), Prod.mk ω ⁻¹' u ×ˢ s ×ˢ t = if ω ∈ u then s ×ˢ t else ∅
ω : α
hω : (κ ω) (f ⁻¹' s ∩ g ⁻¹' t) = (κ ω) (f ⁻¹' s) * (κ ω) (g ⁻¹' t)
hωu : ω ∈ u
⊢ Measurable fun ω ↦ (f ω, g ω)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Independence.Kernel.IndepFun

Goal state before:
pp: α : Type u_1
Ω : Type u_2
β : Type u_4
γ : Type u_6
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteKernel κ
f : Ω → β
g : Ω → γ
hf : Measurable f
hg : Measurable g
h :
  ∀ (s : Set β) (t : Set γ),
    MeasurableSet s → MeasurableSet t → ∀ᵐ (a : α) ∂μ, (κ a) (f ⁻¹' s ∩ g ⁻¹' t) = (κ a) (f ⁻¹' s) * (κ a) (g ⁻¹' t)
u : Set α
s : Set β
t : Set γ
hu : MeasurableSet u
hs : MeasurableSet s
ht : MeasurableSet t
h_set_eq : ∀ (ω : α), Prod.mk ω ⁻¹' u ×ˢ s ×ˢ t = if ω ∈ u then s ×ˢ t else ∅
ω : α
hω : (κ ω) (f ⁻¹' s ∩ g ⁻¹' t) = (κ ω) (f ⁻¹' s) * (κ ω) (g ⁻¹' t)
hωu : ω ∈ u
⊢ Measurable f

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Independence.Kernel.IndepFun

Goal state before:
pp: α : Type u_1
Ω : Type u_2
β : Type u_4
γ : Type u_6
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteKernel κ
f : Ω → β
g : Ω → γ
hf : Measurable f
hg : Measurable g
h :
  ∀ (s : Set β) (t : Set γ),
    MeasurableSet s → MeasurableSet t → ∀ᵐ (a : α) ∂μ, (κ a) (f ⁻¹' s ∩ g ⁻¹' t) = (κ a) (f ⁻¹' s) * (κ a) (g ⁻¹' t)
u : Set α
s : Set β
t : Set γ
hu : MeasurableSet u
hs : MeasurableSet s
ht : MeasurableSet t
h_set_eq : ∀ (ω : α), Prod.mk ω ⁻¹' u ×ˢ s ×ˢ t = if ω ∈ u then s ×ˢ t else ∅
ω : α
hω : (κ ω) (f ⁻¹' s ∩ g ⁻¹' t) = (κ ω) (f ⁻¹' s) * (κ ω) (g ⁻¹' t)
hωu : ω ∈ u
⊢ Measurable g

Tactic:
by fun_prop

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

Goal state before:
pp: case h
Ω : Type u_1
ι : Type u_2
mΩ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : ProbabilityMeasure Ω
μs : ι → ProbabilityMeasure Ω
L : Filter ι
inst✝ : L.IsCountablyGenerated
h₁ : IsTightMeasureSet (range (ProbabilityMeasure.toMeasure ∘ μs))
h₂ : ∀ (F : Set Ω), IsCompact F → limsup (fun x ↦ (μs x) F) L ≤ μ F
h✝ : L.NeBot
F : Set Ω
hF_closed : IsClosed[inst✝²] F
ε : ℝ≥0
hε : 0 < ε
x✝ : ↑(μ F) < ∞
K : Set Ω
hKc : IsCompact K
hK_le : ∀ μ ∈ range (ProbabilityMeasure.toMeasure ∘ μs), μ Kᶜ ≤ ↑ε
i : ι
⊢ ↑(μs i) F ≤ ↑(μs i) (F ∩ K) + ↑ε

Tactic:
grw [measure_mono (t := (F ∩ K) ∪ F \ K) (by simp), measure_union_le]

Goal state after:
case h
Ω : Type u_1
ι : Type u_2
mΩ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : ProbabilityMeasure Ω
μs : ι → ProbabilityMeasure Ω
L : Filter ι
inst✝ : L.IsCountablyGenerated
h₁ : IsTightMeasureSet (range (ProbabilityMeasure.toMeasure ∘ μs))
h₂ : ∀ (F : Set Ω), IsCompact F → limsup (fun x ↦ (μs x) F) L ≤ μ F
h✝ : L.NeBot
F : Set Ω
hF_closed : IsClosed[inst✝²] F
ε : ℝ≥0
hε : 0 < ε
x✝ : ↑(μ F) < ∞
K : Set Ω
hKc : IsCompact K
hK_le : ∀ μ ∈ range (ProbabilityMeasure.toMeasure ∘ μs), μ Kᶜ ≤ ↑ε
i : ι
⊢ ↑(μs i) (F ∩ K) + ↑(μs i) (F \ K) ≤ ↑(μs i) (F ∩ K) + ↑ε
Module: Mathlib.Probability.Independence.Kernel.IndepFun

Goal state before:
pp: α : Type u_1
Ω : Type u_2
β : Type u_4
γ : Type u_6
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteKernel κ
f : Ω → β
g : Ω → γ
hf : Measurable f
hg : Measurable g
s : Set β
t : Set γ
hs : MeasurableSet s
ht : MeasurableSet t
u : Set α
h : ∫⁻ (a : α) in u, ((κ.map fun ω ↦ (f ω, g ω)) a) (s ×ˢ t) ∂μ = ∫⁻ (a : α) in u, ((κ.map f ×ₖ κ.map g) a) (s ×ˢ t) ∂μ
hu : MeasurableSet u
hμu : μ u < ⊤
ω : α
⊢ Measurable fun ω ↦ (f ω, g ω)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Independence.Kernel.IndepFun

Goal state before:
pp: α : Type u_1
Ω : Type u_2
β : Type u_4
γ : Type u_6
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteKernel κ
f : Ω → β
g : Ω → γ
hf : Measurable f
hg : Measurable g
s : Set β
t : Set γ
hs : MeasurableSet s
ht : MeasurableSet t
u : Set α
h : ∫⁻ (a : α) in u, ((κ.map fun ω ↦ (f ω, g ω)) a) (s ×ˢ t) ∂μ = ∫⁻ (a : α) in u, ((κ.map f ×ₖ κ.map g) a) (s ×ˢ t) ∂μ
hu : MeasurableSet u
hμu : μ u < ⊤
ω : α
⊢ Measurable f

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Independence.Kernel.IndepFun

Goal state before:
pp: α : Type u_1
Ω : Type u_2
β : Type u_4
γ : Type u_6
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteKernel κ
f : Ω → β
g : Ω → γ
hf : Measurable f
hg : Measurable g
s : Set β
t : Set γ
hs : MeasurableSet s
ht : MeasurableSet t
u : Set α
h : ∫⁻ (a : α) in u, ((κ.map fun ω ↦ (f ω, g ω)) a) (s ×ˢ t) ∂μ = ∫⁻ (a : α) in u, ((κ.map f ×ₖ κ.map g) a) (s ×ˢ t) ∂μ
hu : MeasurableSet u
hμu : μ u < ⊤
ω : α
⊢ Measurable g

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Density

Goal state before:
pp: Ω : Type u_1
G : Type u_2
mΩ : MeasurableSpace Ω
ℙ : Measure Ω
inst✝⁶ : Group G
mG : MeasurableSpace G
inst✝⁵ : MeasurableMul₂ G
inst✝⁴ : MeasurableInv G
μ : Measure G
inst✝³ : μ.IsMulLeftInvariant
X Y : Ω → G
inst✝² : SFinite μ
inst✝¹ : HasPDF X ℙ μ
inst✝ : HasPDF Y ℙ μ
σX : SigmaFinite (map X ℙ)
σY : SigmaFinite (map Y ℙ)
hXY : X ⟂ᵢ[ℙ] Y
this✝ : AEMeasurable X ℙ
this : AEMeasurable Y ℙ
⊢ AEMeasurable (X * Y) ℙ

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Density

Goal state before:
pp: Ω : Type u_1
G : Type u_2
mΩ : MeasurableSpace Ω
ℙ : Measure Ω
inst✝⁶ : Group G
mG : MeasurableSpace G
inst✝⁵ : MeasurableMul₂ G
inst✝⁴ : MeasurableInv G
μ : Measure G
inst✝³ : μ.IsMulLeftInvariant
X Y : Ω → G
inst✝² : SFinite μ
inst✝¹ : HasPDF X ℙ μ
inst✝ : HasPDF Y ℙ μ
σX : SigmaFinite (map X ℙ)
σY : SigmaFinite (map Y ℙ)
hXY : X ⟂ᵢ[ℙ] Y
this✝ : AEMeasurable X ℙ
this : AEMeasurable Y ℙ
⊢ AEMeasurable X ℙ

Tactic:
by fun_prop

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

Goal state before:
pp: case inl
Ω : Type u_1
ι : Type u_2
inst✝ : MeasurableSpace Ω
S : Set (Set Ω)
hS : IsPiSystem S
μ : ι → Measure Ω
ν : Measure Ω
l : Filter ι
t : Finset (Set Ω)
ht : ∀ s ∈ t, s ∈ S
hmeas : ∀ s ∈ S, MeasurableSet s
h : ∀ s ∈ S, Tendsto (fun i ↦ (μ i).real s) l (𝓝 (ν.real s))
hν : ∀ s ∈ S, ν s ≠ ∞
hμ : ∀ s ∈ S, ∀ (i : ι), (μ i) s ≠ ∞
A : ∀ (i : ι), (μ i).real (⋃ s ∈ t, s) = ∑ u ∈ t.powerset with u.Nonempty, (-1) ^ (#u + 1) * (μ i).real (⋂ s ∈ u, s)
u : Finset (Set Ω)
hu : u ⊆ t ∧ u.Nonempty
h'u : ⋂ s ∈ u, s = ∅
⊢ Tendsto (fun x ↦ (μ x).real (⋂ s ∈ u, s)) l (𝓝 (ν.real (⋂ i ∈ u, i)))

Tactic:
simpa [h'u] using tendsto_const_nhds

Goal state after:
Module: Mathlib.Probability.Density

Goal state before:
pp: Ω : Type u_1
G : Type u_2
mΩ : MeasurableSpace Ω
ℙ : Measure Ω
inst✝⁶ : Group G
mG : MeasurableSpace G
inst✝⁵ : MeasurableMul₂ G
inst✝⁴ : MeasurableInv G
μ : Measure G
inst✝³ : μ.IsMulLeftInvariant
X Y : Ω → G
inst✝² : SFinite μ
inst✝¹ : HasPDF X ℙ μ
inst✝ : HasPDF Y ℙ μ
σX : SigmaFinite (map X ℙ)
σY : SigmaFinite (map Y ℙ)
hXY : X ⟂ᵢ[ℙ] Y
this✝ : AEMeasurable X ℙ
this : AEMeasurable Y ℙ
⊢ AEMeasurable Y ℙ

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Moments.Variance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
hXm : AEStronglyMeasurable X μ
hX : ¬MemLp X 2 μ
h : eVar[X; μ] < ∞
⊢ AEStronglyMeasurable (fun ω ↦ X ω - ∫ (x : Ω), X x ∂μ) μ

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Moments.Covariance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : Integrable X μ
c : ℝ
ω : Ω
⊢ Integrable (fun x ↦ c) μ

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Moments.Variance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : AEStronglyMeasurable X μ
c : ℝ
hX_Lp : MemLp X 2 μ
hX_int : Integrable X μ
⊢ Integrable (fun x ↦ c) μ

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Moments.Variance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : AEStronglyMeasurable X μ
c : ℝ
⊢ AEStronglyMeasurable (fun ω ↦ -X ω) μ

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Independence.Integration

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_5
F : Type u_6
G : Type u_7
inst✝⁹ : TopologicalSpace E
inst✝⁸ : ContinuousENorm E
inst✝⁷ : MeasurableSpace E
inst✝⁶ : OpensMeasurableSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : ContinuousENorm F
inst✝³ : MeasurableSpace F
inst✝² : OpensMeasurableSpace F
inst✝¹ : TopologicalSpace G
inst✝ : ContinuousENorm G
X : Ω → E
Y : Ω → F
hXY : X ⟂ᵢ[μ] Y
hX : Integrable X μ
hY : Integrable Y μ
B : E → F → G
cB : Continuous (Function.uncurry B)
C : ℝ≥0
hB : ∀ (x : E) (y : F), ‖B x y‖ₑ ≤ ↑C * ‖x‖ₑ * ‖y‖ₑ
⊢ AEMeasurable (fun ω ↦ ‖X ω‖ₑ * ‖Y ω‖ₑ) μ

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Independence.Integration

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_5
F : Type u_6
G : Type u_7
𝕜 : Type u_8
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : SeminormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : MeasurableSpace E
inst✝⁶ : OpensMeasurableSpace E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : MeasurableSpace F
inst✝² : OpensMeasurableSpace F
inst✝¹ : SeminormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
X : Ω → E
Y : Ω → F
hXY : X ⟂ᵢ[μ] Y
hX : Integrable X μ
hY : Integrable Y μ
B : E →L[𝕜] F →L[𝕜] G
⊢ Continuous (Function.uncurry fun x1 x2 ↦ (B x1) x2)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Independence.Integration

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_5
F : Type u_6
G : Type u_7
inst✝⁸ : TopologicalSpace E
inst✝⁷ : ContinuousENorm E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : OpensMeasurableSpace E
inst✝⁴ : NormedAddGroup F
inst✝³ : MeasurableSpace F
inst✝² : OpensMeasurableSpace F
inst✝¹ : TopologicalSpace G
inst✝ : ContinuousENorm G
X : Ω → E
Y : Ω → F
hXY : X ⟂ᵢ[μ] Y
B : E → F → G
c : ℝ≥0
hc : c ≠ 0
hB : ∀ (x : E) (y : F), ↑c * ‖x‖ₑ * ‖y‖ₑ ≤ ‖B x y‖ₑ
h'XY : Integrable (fun ω ↦ B (X ω) (Y ω)) μ
hX : AEStronglyMeasurable X μ
hY : AEStronglyMeasurable Y μ
h'Y : ¬Y =ᵐ[μ] 0
I : ∫⁻ (ω : Ω), ‖Y ω‖ₑ ∂μ ≠ 0
H : ∫⁻ (a : Ω), ‖X a‖ₑ ∂μ = ∞
J : (fun x ↦ ‖X x‖ₑ) ⟂ᵢ[μ] fun x ↦ ‖Y x‖ₑ
⊢ AEMeasurable (fun ω ↦ ‖X ω‖ₑ * ‖Y ω‖ₑ) μ

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Moments.Variance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X : Ω → ℝ
Ω' : Type u_3
mΩ' : MeasurableSpace Ω'
μ : Measure Ω'
Y : Ω' → Ω
hX : AEMeasurable X (Measure.map Y μ)
hY : AEMeasurable Y μ
⊢ AEMeasurable (fun ω ↦ X ω - ∫ (x : Ω), X x ∂Measure.map Y μ) (Measure.map Y μ)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Moments.Variance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
X : Ω → ℝ
hm : AEStronglyMeasurable X μ
hX : ¬MemLp X 2 μ
hint : Integrable X μ
h : Integrable (fun a ↦ (X a - ∫ (x : Ω), X x ∂μ) ^ 2) μ
⊢ AEStronglyMeasurable (X - fun ω ↦ ∫ (x : Ω), X x ∂μ) μ

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Independence.Integration

Goal state before:
pp: Ω : Type u_1
𝕜 : Type u_2
inst✝¹⁴ : RCLike 𝕜
mΩ : MeasurableSpace Ω
μ : Measure Ω
𝓧 : Type u_3
𝓨 : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
inst✝¹³ : MeasurableSpace 𝓧
inst✝¹² : MeasurableSpace 𝓨
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : CompleteSpace E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : CompleteSpace F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : CompleteSpace G
X : Ω → 𝓧
Y : Ω → 𝓨
f : 𝓧 → E
g : 𝓨 → F
hXY : X ⟂ᵢ[μ] Y
hX : AEMeasurable X μ
hY : AEMeasurable Y μ
hf : Integrable f (Measure.map X μ)
hg : Integrable g (Measure.map Y μ)
B : E →L[𝕜] F →L[𝕜] G
h : ¬∀ᵐ (ω : Ω) ∂μ, f (X ω) = 0
this✝³ : MeasurableSpace E := borel E
this✝² : BorelSpace E
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
this : IsProbabilityMeasure μ
⊢ AEMeasurable (fun ω ↦ (X ω, Y ω)) μ

Tactic:
by fun_prop

Goal state after: