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

Goal state before:
pp: case f_bdd
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
ε : ℝ
ε_pos : 0 < ε
hμν : levyProkhorovEDist μ ν < ENNReal.ofReal ε
f : Ω →ᵇ ℝ
f_nn : 0 ≤ᵐ[μ] ⇑f
key : (fun t ↦ μ.real {a | t ≤ f a}) ≤ fun t ↦ ν.real (thickening ε {a | t ≤ f a}) + ε
intble₁ : IntegrableOn (fun t ↦ μ.real {a | t ≤ f a}) (Ioc 0 ‖f‖) volume
⊢ ∀ᵐ (a : ℝ) ∂volume.restrict (Ioc 0 ‖f‖), ‖ν.real (thickening ε {a_1 | a ≤ f a_1})‖ ≤ ν.real univ

Tactic:
apply Eventually.of_forall <| fun t ↦ ?_

Goal state after:
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
ε : ℝ
ε_pos : 0 < ε
hμν : levyProkhorovEDist μ ν < ENNReal.ofReal ε
f : Ω →ᵇ ℝ
f_nn : 0 ≤ᵐ[μ] ⇑f
key : (fun t ↦ μ.real {a | t ≤ f a}) ≤ fun t ↦ ν.real (thickening ε {a | t ≤ f a}) + ε
intble₁ : IntegrableOn (fun t ↦ μ.real {a | t ≤ f a}) (Ioc 0 ‖f‖) volume
t : ℝ
⊢ ‖ν.real (thickening ε {a | t ≤ f a})‖ ≤ ν.real univ
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: case refine_1
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : MeasurableSpace X
inst✝¹ : BorelSpace X
μ : Measure X
inst✝ : IsFiniteMeasure μ
K : Set X
hK : IsCompact K
h : μ Kᶜ = 0
μ' : Measure ↑K := Measure.comap Subtype.val μ
ν' : Measure ↑K
ν'_reg : ν'.Regular
ν'_fin : IsFiniteMeasure ν'
hν' : ∀ (g : ↑K →ᵇ ℝ), ∫ (x : ↑K), g x ∂μ' = ∫ (x : ↑K), g x ∂ν'
⊢ (Measure.map Subtype.val ν') Kᶜ = 0

Tactic:
rw [Measure.map_apply (by fun_prop) hK.measurableSet.compl]
    simp

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.Extension

Goal state before:
pp: A : Type u_1
B : Type u_2
C : Type u_3
E : Type u_4
inst✝¹³ : Group A
inst✝¹² : Group B
inst✝¹¹ : Group C
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : TopologicalSpace B
inst✝⁸ : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst✝⁷ : IsTopologicalGroup A
inst✝⁶ : IsTopologicalGroup B
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : MeasurableSpace A
inst✝³ : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst✝² : NormedSpace ℝ E
inst✝¹ : IsTopologicalGroup C
inst✝ : LocallyCompactSpace B
f : CompactlySupportedContinuousMap B E
this : UniformSpace B := IsTopologicalGroup.rightUniformSpace B
b : B
ε : ℝ
hε : ε > 0
U₀ : Set B
hU₀ : IsCompact U₀
hb : U₀ ∈ nhds b
K : Set B
hK : IsCompact K
hf₀ : ∀ x ∉ K, f x = 0
S : Set A := ⇑φ ⁻¹' (U₀⁻¹ * K)
hSc : IsCompact S
δ : ℝ
hδ0 : δ > 0
hδ : ENNReal.ofReal δ * μA S < ENNReal.ofReal ε
hS : ∀ {x : B}, x ∈ U₀ → ∀ {y : A}, y ∉ S → (H.pullback f x) y = 0
ha : UniformContinuous ⇑f
⊢ ∀ᶠ (x : B) in nhds b, dist (∫ (a : A), (H.pullback f x) a ∂μA) (∫ (a : A), (H.pullback f b) a ∂μA) < ε

Tactic:
rw [uniformContinuous_iff_eventually] at ha

Goal state after:
A : Type u_1
B : Type u_2
C : Type u_3
E : Type u_4
inst✝¹³ : Group A
inst✝¹² : Group B
inst✝¹¹ : Group C
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : TopologicalSpace B
inst✝⁸ : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst✝⁷ : IsTopologicalGroup A
inst✝⁶ : IsTopologicalGroup B
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : MeasurableSpace A
inst✝³ : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst✝² : NormedSpace ℝ E
inst✝¹ : IsTopologicalGroup C
inst✝ : LocallyCompactSpace B
f : CompactlySupportedContinuousMap B E
this : UniformSpace B := IsTopologicalGroup.rightUniformSpace B
b : B
ε : ℝ
hε : ε > 0
U₀ : Set B
hU₀ : IsCompact U₀
hb : U₀ ∈ nhds b
K : Set B
hK : IsCompact K
hf₀ : ∀ x ∉ K, f x = 0
S : Set A := ⇑φ ⁻¹' (U₀⁻¹ * K)
hSc : IsCompact S
δ : ℝ
hδ0 : δ > 0
hδ : ENNReal.ofReal δ * μA S < ENNReal.ofReal ε
hS : ∀ {x : B}, x ∈ U₀ → ∀ {y : A}, y ∉ S → (H.pullback f x) y = 0
ha : ∀ r ∈ uniformity E, ∀ᶠ (x : B × B) in uniformity B, (f x.1, f x.2) ∈ r
⊢ ∀ᶠ (x : B) in nhds b, dist (∫ (a : A), (H.pullback f x) a ∂μA) (∫ (a : A), (H.pullback f b) a ∂μA) < ε
Module: Mathlib.MeasureTheory.Measure.Haar.Extension

Goal state before:
pp: A : Type u_1
B : Type u_2
C : Type u_3
E : Type u_4
inst✝¹³ : Group A
inst✝¹² : Group B
inst✝¹¹ : Group C
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : TopologicalSpace B
inst✝⁸ : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst✝⁷ : IsTopologicalGroup A
inst✝⁶ : IsTopologicalGroup B
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : MeasurableSpace A
inst✝³ : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst✝² : NormedSpace ℝ E
inst✝¹ : IsTopologicalGroup C
inst✝ : LocallyCompactSpace B
f : CompactlySupportedContinuousMap B E
this : UniformSpace B := IsTopologicalGroup.rightUniformSpace B
ε : ℝ
hε : ε > 0
U₀ : Set B
hU₀ : IsCompact U₀
K : Set B
hK : IsCompact K
hf₀ : ∀ x ∉ K, f x = 0
S : Set A := ⇑φ ⁻¹' (U₀⁻¹ * K)
hSc : IsCompact S
δ : ℝ
hδ0 : δ > 0
hδ : ENNReal.ofReal δ * μA S < ENNReal.ofReal ε
hS : ∀ {x : B}, x ∈ U₀ → ∀ {y : A}, y ∉ S → (H.pullback f x) y = 0
ha : ∀ r ∈ uniformity E, ∀ᶠ (x : B × B) in uniformity B, (f x.1, f x.2) ∈ r
U : Set B
hU : U ∈ nhds 1
hf : (fun p ↦ p.2 * p.1⁻¹) ⁻¹' U ⊆ {x | (fun x ↦ (f x.1, f x.2) ∈ {p | dist p.1 p.2 < δ}) x}
t : B
ht : t ∈ U⁻¹
b : B
hb : U₀ ∈ nhds b
htb : (fun x1 x2 ↦ x1 * x2) t b ∈ U₀
h : ∀ a ∈ S, edist ((H.pullback f (t * b)) a) ((H.pullback f b) a) ≤ ENNReal.ofReal δ
y : A
hy : y ∉ S
⊢ y ∉ Function.support fun a ↦ edist ((H.pullback f (t * b)) a) ((H.pullback f b) a)

Tactic:
Function.notMem_support,

Goal state after:
A : Type u_1
B : Type u_2
C : Type u_3
E : Type u_4
inst✝¹³ : Group A
inst✝¹² : Group B
inst✝¹¹ : Group C
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : TopologicalSpace B
inst✝⁸ : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst✝⁷ : IsTopologicalGroup A
inst✝⁶ : IsTopologicalGroup B
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : MeasurableSpace A
inst✝³ : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst✝² : NormedSpace ℝ E
inst✝¹ : IsTopologicalGroup C
inst✝ : LocallyCompactSpace B
f : CompactlySupportedContinuousMap B E
this : UniformSpace B := IsTopologicalGroup.rightUniformSpace B
ε : ℝ
hε : ε > 0
U₀ : Set B
hU₀ : IsCompact U₀
K : Set B
hK : IsCompact K
hf₀ : ∀ x ∉ K, f x = 0
S : Set A := ⇑φ ⁻¹' (U₀⁻¹ * K)
hSc : IsCompact S
δ : ℝ
hδ0 : δ > 0
hδ : ENNReal.ofReal δ * μA S < ENNReal.ofReal ε
hS : ∀ {x : B}, x ∈ U₀ → ∀ {y : A}, y ∉ S → (H.pullback f x) y = 0
ha : ∀ r ∈ uniformity E, ∀ᶠ (x : B × B) in uniformity B, (f x.1, f x.2) ∈ r
U : Set B
hU : U ∈ nhds 1
hf : (fun p ↦ p.2 * p.1⁻¹) ⁻¹' U ⊆ {x | (fun x ↦ (f x.1, f x.2) ∈ {p | dist p.1 p.2 < δ}) x}
t : B
ht : t ∈ U⁻¹
b : B
hb : U₀ ∈ nhds b
htb : (fun x1 x2 ↦ x1 * x2) t b ∈ U₀
h : ∀ a ∈ S, edist ((H.pullback f (t * b)) a) ((H.pullback f b) a) ≤ ENNReal.ofReal δ
y : A
hy : y ∉ S
⊢ edist ((H.pullback f (t * b)) y) ((H.pullback f b) y) = 0
Module: Mathlib.MeasureTheory.Measure.IntegralCharFun

Goal state before:
pp: μ : Measure ℝ
r : ℝ
inst✝ : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ
⊢ 2 * ‖∫ (x : ℝ), 1 - sinc (2 * r⁻¹ * x) ∂μ‖ = 2⁻¹ * r * (‖2‖ * ‖2 * r⁻¹ - 2 * r⁻¹ * ∫ (x : ℝ), sinc (2 * r⁻¹ * x) ∂μ‖)

Tactic:
Real.norm_ofNat,

Goal state after:
μ : Measure ℝ
r : ℝ
inst✝ : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ
⊢ 2 * ‖∫ (x : ℝ), 1 - sinc (2 * r⁻¹ * x) ∂μ‖ = 2⁻¹ * r * (2 * ‖2 * r⁻¹ - 2 * r⁻¹ * ∫ (x : ℝ), sinc (2 * r⁻¹ * x) ∂μ‖)
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: case h₁
E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
C : ℝ≥0
K : Set E
hK : IsCompact K
f : ↑K → E := Subtype.val
hf : IsClosedEmbedding f
rf : range f = K
F : FiniteMeasure ↑K → FiniteMeasure E := fun μ ↦ μ.map f
T : Set (FiniteMeasure ↑K) := {μ | μ.mass ≤ C}
μ : FiniteMeasure E
hμ : μ ∈ {μ | μ.mass ≤ C ∧ μ Kᶜ = 0}
s : Set E
hs : MeasurableSet s
⊢ (Measure.comap f ↑μ) (Subtype.val ⁻¹' s) = ↑μ s

Tactic:
Measure.comap_apply _ (Subtype.val_injective),

Goal state after:
case h₁
E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
C : ℝ≥0
K : Set E
hK : IsCompact K
f : ↑K → E := Subtype.val
hf : IsClosedEmbedding f
rf : range f = K
F : FiniteMeasure ↑K → FiniteMeasure E := fun μ ↦ μ.map f
T : Set (FiniteMeasure ↑K) := {μ | μ.mass ≤ C}
μ : FiniteMeasure E
hμ : μ ∈ {μ | μ.mass ≤ C ∧ μ Kᶜ = 0}
s : Set E
hs : MeasurableSet s
⊢ ↑μ (Subtype.val '' Subtype.val ⁻¹' s) = ↑μ s

case h₁.hf
E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
C : ℝ≥0
K : Set E
hK : IsCompact K
f : ↑K → E := Subtype.val
hf : IsClosedEmbedding f
rf : range f = K
F : FiniteMeasure ↑K → FiniteMeasure E := fun μ ↦ μ.map f
T : Set (FiniteMeasure ↑K) := {μ | μ.mass ≤ C}
μ : FiniteMeasure E
hμ : μ ∈ {μ | μ.mass ≤ C ∧ μ Kᶜ = 0}
s : Set E
hs : MeasurableSet s
⊢ ∀ (s : Set { x // x ∈ K }), MeasurableSet s → MeasurableSet (Subtype.val '' s)

case h₁.hs
E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
C : ℝ≥0
K : Set E
hK : IsCompact K
f : ↑K → E := Subtype.val
hf : IsClosedEmbedding f
rf : range f = K
F : FiniteMeasure ↑K → FiniteMeasure E := fun μ ↦ μ.map f
T : Set (FiniteMeasure ↑K) := {μ | μ.mass ≤ C}
μ : FiniteMeasure E
hμ : μ ∈ {μ | μ.mass ≤ C ∧ μ Kᶜ = 0}
s : Set E
hs : MeasurableSet s
⊢ MeasurableSet (Subtype.val ⁻¹' s)
Module: Mathlib.MeasureTheory.Measure.TightNormed

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
inst✝³ : NormedAddCommGroup E
inst✝² : BorelSpace E
inst✝¹ : ProperSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖x‖}) atTop) atTop (𝓝 0)
⊢ Tendsto (fun r ↦ ⨆ μ_1 ∈ Set.range μ, μ_1 {x | r < ‖x‖}) atTop (𝓝 0)

Tactic:
iSup_range

Goal state after:
E : Type u_1
mE : MeasurableSpace E
inst✝³ : NormedAddCommGroup E
inst✝² : BorelSpace E
inst✝¹ : ProperSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖x‖}) atTop) atTop (𝓝 0)
⊢ Tendsto (fun r ↦ ⨆ i, (μ i) {x | r < ‖x‖}) atTop (𝓝 0)
Module: Mathlib.MeasureTheory.Measure.TightNormed

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
inst✝³ : NormedAddCommGroup E
inst✝² : BorelSpace E
inst✝¹ : ProperSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖x‖}) atTop) atTop (𝓝 0)
⊢ IsTightMeasureSet (Set.range μ)

Tactic:
by
  simp_rw [isTightMeasureSet_iff_tendsto_measure_norm_gt, iSup_range]
  refine Nat.tendsto_iSup_of_tendsto_limsup (fun n ↦ ?_) h (fun n u v huv ↦ by gcongr)
  have h_tight : IsTightMeasureSet {μ n} := isTightMeasureSet_singleton
  rw [isTightMeasureSet_iff_tendsto_measure_norm_gt] at h_tight
  simpa using h_tight

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

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
inst✝³ : NormedAddCommGroup E
inst✝² : BorelSpace E
inst✝¹ : ProperSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : IsTightMeasureSet (Set.range μ)
h_sup : Tendsto (fun r ↦ ⨆ μ_1 ∈ Set.range μ, μ_1 {x | r < ‖x‖}) atTop (𝓝 0)
r : ℝ
⊢ limsup (fun n ↦ (μ n) {x | r < ‖x‖}) atTop ≤ ⨆ μ_1 ∈ Set.range μ, μ_1 {x | r < ‖x‖}

Tactic:
iSup_range

Goal state after:
E : Type u_1
mE : MeasurableSpace E
inst✝³ : NormedAddCommGroup E
inst✝² : BorelSpace E
inst✝¹ : ProperSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : IsTightMeasureSet (Set.range μ)
h_sup : Tendsto (fun r ↦ ⨆ μ_1 ∈ Set.range μ, μ_1 {x | r < ‖x‖}) atTop (𝓝 0)
r : ℝ
⊢ limsup (fun n ↦ (μ n) {x | r < ‖x‖}) atTop ≤ ⨆ n, (μ n) {x | r < ‖x‖}
Module: Mathlib.MeasureTheory.Measure.TightNormed

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
S : Set (Measure E)
inst✝³ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
h : Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖x‖}) atTop (𝓝 0)
y : E
this : ProperSpace E
hy : ¬y = 0
h' : Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r * ‖y‖⁻¹ < ‖x‖}) atTop (𝓝 0)
r : ℝ
h_le : ∀ (μ : Measure E), μ {x | r < ‖⟪y, x⟫_𝕜‖} ≤ μ {x | r * ‖y‖⁻¹ < ‖x‖}
μ : Measure E
hμS : μ ∈ S
⊢ μ {x | r < ‖⟪y, x⟫_𝕜‖} ≤ ⨆ (_ : μ ∈ S), μ {x | r * ‖y‖⁻¹ < ‖x‖}

Tactic:
simp [hμS, h_le]

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

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
z : E
⊢ Tendsto (fun r ↦ ⨆ μ_1 ∈ Set.range μ, μ_1 {x | r < ‖⟪z, x⟫_𝕜‖}) atTop (𝓝 0)

Tactic:
iSup_range

Goal state after:
E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
z : E
⊢ Tendsto (fun r ↦ ⨆ i, (μ i) {x | r < ‖⟪z, x⟫_𝕜‖}) atTop (𝓝 0)
Module: Mathlib.MeasureTheory.Measure.TightNormed

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), Tendsto (fun r ↦ ⨆ μ_1 ∈ Set.range μ, μ_1 {x | r < ‖⟪y, x⟫_𝕜‖}) atTop (𝓝 0)
z : E
r : ℝ
⊢ limsup (fun n ↦ (μ n) {x | r < ‖⟪z, x⟫_𝕜‖}) atTop ≤ ⨆ μ_1 ∈ Set.range μ, μ_1 {x | r < ‖⟪z, x⟫_𝕜‖}

Tactic:
iSup_range

Goal state after:
E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), Tendsto (fun r ↦ ⨆ μ_1 ∈ Set.range μ, μ_1 {x | r < ‖⟪y, x⟫_𝕜‖}) atTop (𝓝 0)
z : E
r : ℝ
⊢ limsup (fun n ↦ (μ n) {x | r < ‖⟪z, x⟫_𝕜‖}) atTop ≤ ⨆ n, (μ n) {x | r < ‖⟪z, x⟫_𝕜‖}
Module: Mathlib.MeasureTheory.Measure.Support

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : MeasurableSpace X
μ : Measure X
⊢ μ.support = {x | ∀ (u : Set X), x ∈ u → IsOpen[inst✝¹] u → 0 < μ u}

Tactic:
simp [Set.ext_iff, nhds_basis_opens _ |>.mem_measureSupport]

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

Goal state before:
pp: X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
h𝒜 : μ.MeasureDense 𝒜
t : Set X
ht : t ∈ 𝒜
hμt : μ (∅ ∆ t) < ENNReal.ofReal 1
⊢ t = ∅ ∆ t

Tactic:
← bot_eq_empty,

Goal state after:
X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
h𝒜 : μ.MeasureDense 𝒜
t : Set X
ht : t ∈ 𝒜
hμt : μ (∅ ∆ t) < ENNReal.ofReal 1
⊢ t = ⊥ ∆ t
Module: Mathlib.MeasureTheory.Measure.LevyConvergence

Goal state before:
pp: case hf
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
ι : Type u_2
𝓕 : Filter ι
μ₀ : ProbabilityMeasure E
μ✝ : ι → ProbabilityMeasure E
h : ∀ (t : E), Tendsto (fun n ↦ charFun (↑(μ✝ n)) t) 𝓕 (𝓝 (charFun (↑μ₀) t))
g : E →ᵇ ℂ
w : AddMonoidAlgebra ℂ E
hw : ⇑g = fun x ↦ w.coeff.sum fun a z ↦ z * ↑(probChar (((innerₗ E) x) a))
μ : Measure E
hμ : IsProbabilityMeasure μ
i : E
hi : i ∈ w.coeff.support
⊢ Integrable (fun x ↦ ↑(probChar (((innerₗ E) x) i))) μ

Tactic:
change Integrable (innerProbChar i) μ

Goal state after:
case hf
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
ι : Type u_2
𝓕 : Filter ι
μ₀ : ProbabilityMeasure E
μ✝ : ι → ProbabilityMeasure E
h : ∀ (t : E), Tendsto (fun n ↦ charFun (↑(μ✝ n)) t) 𝓕 (𝓝 (charFun (↑μ₀) t))
g : E →ᵇ ℂ
w : AddMonoidAlgebra ℂ E
hw : ⇑g = fun x ↦ w.coeff.sum fun a z ↦ z * ↑(probChar (((innerₗ E) x) a))
μ : Measure E
hμ : IsProbabilityMeasure μ
i : E
hi : i ∈ w.coeff.support
⊢ Integrable (⇑(innerProbChar i)) μ
Module: Mathlib.MeasureTheory.Measure.SubFinite

Goal state before:
pp: case refine_1
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
inst✝ : IsFiniteMeasure (μ.withDensity g)
hf : Measurable f
hg : Measurable g
⊢ μ.withDensity (f - g) ≤ μ.withDensity f - μ.withDensity g

Tactic:
let t := {x | f x ≤ g x}

Goal state after:
case refine_1
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
inst✝ : IsFiniteMeasure (μ.withDensity g)
hf : Measurable f
hg : Measurable g
t : Set α := {x | f x ≤ g x}
⊢ μ.withDensity (f - g) ≤ μ.withDensity f - μ.withDensity g
Module: Mathlib.MeasureTheory.Measure.SeparableMeasure

Goal state before:
pp: X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
h𝒜 : IsSetAlgebra 𝒜
S : μ.FiniteSpanningSetsIn 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
ms : MeasurableSet s
hμs : μ s ≠ ∞
ε : ℝ
ε_pos : 0 < ε
T : ℕ → Set X := accumulate S.set
T_mem : ∀ (n : ℕ), T n ∈ 𝒜
T_finite : ∀ (n : ℕ), μ (T n) < ∞
T_spanning : ⋃ n, T n = univ
mono : Monotone fun n ↦ T n ∩ s
this✝ : Filter.Tendsto (fun n ↦ ((⇑μ ∘ fun n ↦ T n ∩ s) n).toReal) Filter.atTop (nhds (μ (⋃ n, T n ∩ s)).toReal)
N : ℕ
hN : ∀ n ≥ N, dist ((⇑μ ∘ fun n ↦ T n ∩ s) n).toReal (μ (⋃ n, T n ∩ s)).toReal < ε / 2
this : Fact (μ (T N) < ∞)
t : Set X
t_mem : t ∈ 𝒜
ht : (μ.restrict (T N)) (s ∆ t) < ENNReal.ofReal (ε / 2)
⊢ μ (s ∆ (t ∩ T N)) ≤ μ (s ∆ (s ∩ T N)) + μ ((s ∩ T N) ∆ (t ∩ T N))

Tactic:
exact measure_symmDiff_le _ _ _

Goal state after:
Module: Mathlib.MeasureTheory.Order.UpperLower

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x : ι → ℝ
f : (δ : ℝ) → 0 < δ → ι → ℝ
hf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ
hf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior s
⊢ ¬Tendsto (fun r ↦ volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 0)

Tactic:
intro H

Goal state after:
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x : ι → ℝ
f : (δ : ℝ) → 0 < δ → ι → ℝ
hf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ
hf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior s
H : Tendsto (fun r ↦ volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 0)
⊢ False
Module: Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne

Goal state before:
pp: case mpr
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝² : IsZeroOneMeasure μ
inst✝¹ : StandardBorelSpace α
inst✝ : NeZero μ
this : IsProbabilityMeasure μ
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAsep : ∀ x ∈ univ, ∀ y ∈ univ, (∀ (n : ℕ), x ∈ A n ↔ y ∈ A n) → x = y
B : ℕ → Set α := ⋯
mBn : MeasurableSet (⋂ n, B n)
hBn : μ (⋂ n, B n) = 1
neBn : (⋂ n, B n).Nonempty
y : α
n : ℕ
hsome : ⋯.some ∈ if μ (A n) = 1 then A n else (A n)ᶜ
hy : y ∈ if μ (A n) = 1 then A n else (A n)ᶜ
⊢ neBn.some ∈ A n → y ∈ A n

Tactic:
· intro h
      split_ifs at hsome with hμAn
      · simpa [hμAn] using! hy
      · contradiction

Goal state after:
Module: Mathlib.MeasureTheory.Order.UpperLower

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x : ι → ℝ
f : (δ : ℝ) → 0 < δ → ι → ℝ
hf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ
hf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior sᶜ
⊢ ¬Tendsto (fun r ↦ volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 1)

Tactic:
intro H

Goal state after:
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x : ι → ℝ
f : (δ : ℝ) → 0 < δ → ι → ℝ
hf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ
hf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior sᶜ
H : Tendsto (fun r ↦ volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0) (𝓝 1)
⊢ False
Module: Mathlib.MeasureTheory.OuterMeasure.OfAddContent

Goal state before:
pp: α : Type u_1
C : Set (Set α)
s : Set α
hC : IsSetSemiring C
m : AddContent ℝ≥0∞ C
m_sigma_subadd : m.IsSigmaSubadditive
m_top : ∀ s ∉ C, m s = ∞
hs : s ∈ C
⊢ m s ≤ ⨅ t, ⨅ (_ : ∀ (i : ℕ), t i ∈ C), ⨅ (_ : s ⊆ ⋃ i, t i), ∑' (i : ℕ), m (t i)

Tactic:
refine le_iInf fun f ↦ le_iInf fun hf ↦ le_iInf fun hs_subset ↦ ?_

Goal state after:
α : Type u_1
C : Set (Set α)
s : Set α
hC : IsSetSemiring C
m : AddContent ℝ≥0∞ C
m_sigma_subadd : m.IsSigmaSubadditive
m_top : ∀ s ∉ C, m s = ∞
hs : s ∈ C
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hs_subset : s ⊆ ⋃ i, f i
⊢ m s ≤ ∑' (i : ℕ), m (f i)
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Basic

Goal state before:
pp: X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝³ : TopologicalSpace V
inst✝² : ENormedAddCommMonoid V
inst✝¹ : T2Space V
μ : VectorMeasure X V
Y : Type u_3
inst✝ : MeasurableSpace Y
φ : X → Y
hφ : MeasurableEmbedding φ
s : Set Y
hs : MeasurableSet s
this : (μ.map φ).variation s = (μ.map φ).variation (s ∩ Set.range φ)
t : Set X
ht : MeasurableSet t
⊢ ‖μ t‖ₑ ≤ ‖(μ.map φ) (φ '' t)‖ₑ

Tactic:
rw [map_apply _ hφ.measurable (hφ.measurableSet_image.2 ht), preimage_image_eq _ hφ.injective]

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Basic

Goal state before:
pp: X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup V
μ ν : VectorMeasure X V
inst✝ : Finite X
⊢ μ.variation Set.univ < ∞

Tactic:
let : Fintype X := Fintype.ofFinite X

Goal state after:
X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup V
μ ν : VectorMeasure X V
inst✝ : Finite X
this : Fintype X := Fintype.ofFinite X
⊢ μ.variation Set.univ < ∞
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: 𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst✝³ : PseudoMetricSpace 𝓧
inst✝² : OpensMeasurableSpace 𝓧
inst✝¹ : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
inst✝ : CompleteSpace 𝓧
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
hnonempty : Nonempty 𝓧
D : ℕ → 𝓧
hD : DenseRange D
u : ℕ → ℝ
hu_anti : StrictAnti u
hu_pos : ∀ (n : ℕ), 0 < u n
hu : Tendsto u atTop (𝓝 0)
hcov : ∀ (m : ℕ), ⋃ i, ball (D i) (u m) = univ
ε : ℝ≥0∞
εpos : 0 < ε
hεbound : ε ≤ 1
km : ℕ → ℕ
hbound : ∀ (m : ℕ), ∀ μ ∈ S, 1 - ε * 2 ^ (-↑m) < ↑(μ (⋃ i, ⋃ (_ : i ≤ km m), ball (D i) (u m)))
bigK : Set 𝓧 :=
  ⋂ m, ⋃ i, ⋃ (_ : i ≤ km (m + 1)), closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (ball (D i) (u m))
μ : ProbabilityMeasure 𝓧
hs : μ ∈ S
⊢ ∑' (m : ℕ), ε * 2 ^ (-(↑m + 1)) = ε

Tactic:
rw [ENNReal.tsum_mul_left]
      nth_rw 2 [← mul_one (a := ε)]
      congr
      ring_nf
      exact tsum_two_zpow_neg_add_one

Goal state after: