mathlib-initiative/mathlib-tactics huggingface dataset:
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) * ‖1 - ∫ (x : ℝ), sinc (2 * r⁻¹ * x) ∂μ‖

Tactic:
congr

Goal state after:
case e_a
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 = 2⁻¹ * r * 2 * (2 * r⁻¹)

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

Goal state before:
pp: E : Type u_1
inst✝⁴ : MeasurableSpace E
inst✝³ : TopologicalSpace E
inst✝² : T2Space E
inst✝¹ : BorelSpace E
inst✝ : CompactSpace E
C : ℝ≥0
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C} ∈ f
g : E →C_c ℝ
μ : FiniteMeasure E
hμ : μ.mass ≤ C
⊢ -(↑C * ‖g.toBoundedContinuousFunction‖) ≤ ∫ (x : E), g x ∂↑μ ∧
    ∫ (x : E), g x ∂↑μ ≤ ↑C * ‖g.toBoundedContinuousFunction‖

Tactic:
refine ⟨?_, ?_⟩

Goal state after:
case refine_1
E : Type u_1
inst✝⁴ : MeasurableSpace E
inst✝³ : TopologicalSpace E
inst✝² : T2Space E
inst✝¹ : BorelSpace E
inst✝ : CompactSpace E
C : ℝ≥0
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C} ∈ f
g : E →C_c ℝ
μ : FiniteMeasure E
hμ : μ.mass ≤ C
⊢ -(↑C * ‖g.toBoundedContinuousFunction‖) ≤ ∫ (x : E), g x ∂↑μ

case refine_2
E : Type u_1
inst✝⁴ : MeasurableSpace E
inst✝³ : TopologicalSpace E
inst✝² : T2Space E
inst✝¹ : BorelSpace E
inst✝ : CompactSpace E
C : ℝ≥0
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C} ∈ f
g : E →C_c ℝ
μ : FiniteMeasure E
hμ : μ.mass ≤ C
⊢ ∫ (x : E), g x ∂↑μ ≤ ↑C * ‖g.toBoundedContinuousFunction‖
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : Nontrivial E
k : ℕ
hk : finrank ℝ E = 2 * k
x : E
r : ℝ
⊢ (ENNReal.ofReal r ^ 2) ^ k * ENNReal.ofReal ((√π ^ 2) ^ k / Gamma (↑(2 * k) / 2 + 1)) =
    (ENNReal.ofReal r ^ 2) ^ k * ENNReal.ofReal (π ^ k / ↑k !)

Tactic:
sq_sqrt pi_nonneg

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : Nontrivial E
k : ℕ
hk : finrank ℝ E = 2 * k
x : E
r : ℝ
⊢ (ENNReal.ofReal r ^ 2) ^ k * ENNReal.ofReal (π ^ k / Gamma (↑(2 * k) / 2 + 1)) =
    (ENNReal.ofReal r ^ 2) ^ k * ENNReal.ofReal (π ^ k / ↑k !)
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : Nontrivial E
k : ℕ
hk : finrank ℝ E = 2 * k
x : E
r : ℝ
⊢ (ENNReal.ofReal r ^ 2) ^ k * ENNReal.ofReal (π ^ k / Gamma (↑(2 * k) / 2 + 1)) =
    (ENNReal.ofReal r ^ 2) ^ k * ENNReal.ofReal (π ^ k / ↑k !)

Tactic:
congr

Goal state after:
case e_a.e_a
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : Nontrivial E
k : ℕ
hk : finrank ℝ E = 2 * k
x : E
r : ℝ
⊢ Gamma (↑(2 * k) / 2 + 1) = ↑k !
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:
simp_rw [isTightMeasureSet_iff_tendsto_measure_norm_gt, 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)
n : ℕ
h_tight : IsTightMeasureSet {μ n}
⊢ Tendsto (fun r ↦ (μ n) {x | r < ‖x‖}) atTop (𝓝 0)

Tactic:
isTightMeasureSet_iff_tendsto_measure_norm_gt

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)
n : ℕ
h_tight : Tendsto (fun r ↦ ⨆ μ_1 ∈ {μ n}, μ_1 {x | r < ‖x‖}) atTop (𝓝 0)
⊢ Tendsto (fun r ↦ (μ n) {x | r < ‖x‖}) atTop (𝓝 0)
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 : IsTightMeasureSet S
y : E
this : ProperSpace E
⊢ Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖⟪y, x⟫_𝕜‖}) atTop (𝓝 0)

Tactic:
isTightMeasureSet_iff_tendsto_measure_norm_gt

Goal state after:
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
⊢ Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖⟪y, 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 ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
z : E
n : ℕ
h_tight : IsTightMeasureSet {Measure.map (fun x ↦ ⟪z, x⟫_𝕜) (μ n)}
⊢ Tendsto (fun r ↦ (μ n) {x | r < ‖⟪z, x⟫_𝕜‖}) atTop (𝓝 0)

Tactic:
isTightMeasureSet_iff_tendsto_measure_norm_gt

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
n : ℕ
h_tight : Tendsto (fun r ↦ ⨆ μ_1 ∈ {Measure.map (fun x ↦ ⟪z, x⟫_𝕜) (μ n)}, μ_1 {x | r < ‖x‖}) atTop (𝓝 0)
⊢ Tendsto (fun r ↦ (μ n) {x | r < ‖⟪z, x⟫_𝕜‖}) atTop (𝓝 0)
Module: Mathlib.MeasureTheory.Measure.MeasuredSets

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hC : IsSetRing C
h'C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0
h : mα = generateFrom C
s : Set α
hs : MeasurableSet s
ε✝ : ℝ≥0∞
hε : 0 < ε✝
ε : ℝ≥0∞
εpos : 0 < ε
⊢ μ (∅ ∆ ∅) < ε

Tactic:
simp [εpos]

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hC : IsSetRing C
h'C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0
h : mα = generateFrom C
s : Set α
hs : MeasurableSet s
ε✝ : ℝ≥0∞
hε : 0 < ε✝
ε : ℝ≥0∞
εpos : 0 < ε
⊢ μ (∅ ∆ ∅) < ε

Tactic:
by simp [εpos]

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hC : IsSetRing C
h'C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0
h : mα = generateFrom C
s✝ : Set α
hs✝ : MeasurableSet s✝
ε✝ : ℝ≥0∞
hε : 0 < ε✝
s : Set α
hs : s ∈ C
ε : ℝ≥0∞
εpos : 0 < ε
⊢ μ (s ∆ s) < ε

Tactic:
simp [εpos]

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hC : IsSetRing C
h'C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0
h : mα = generateFrom C
s✝ : Set α
hs✝ : MeasurableSet s✝
ε✝ : ℝ≥0∞
hε : 0 < ε✝
s : Set α
hs : s ∈ C
ε : ℝ≥0∞
εpos : 0 < ε
⊢ μ (s ∆ s) < ε

Tactic:
by simp [εpos]

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

Goal state before:
pp: X : Type u_1
inst✝ : MeasurableSpace X
f : Set X → ℝ≥0∞
s : ℕ → Set X
hs : ∀ (i : ℕ), MeasurableSet (s i)
hs' : Pairwise (Disjoint on s)
n : ℕ
hn : ¬n = 0
ε' : ℝ≥0
hε' : 0 < ε'
hsnetop : preVariationFun f (⋃ i, s i) < ∞
ε : ℝ≥0 := ε' / ↑n
hε : 0 < ε
hs'' : ∀ (i : ℕ), preVariationFun f (s i) ≠ ∞
P : (i : ℕ) → Finpartition ⟨s i, ⋯⟩
hP : ∀ (i : ℕ), preVariationFun f (s i) ≤ ∑ p ∈ (P i).parts, f ↑p + ↑ε
⊢ ∑ i ∈ Finset.range n, (∑ p ∈ (P i).parts, f ↑p + ↑ε) = ∑ i ∈ Finset.range n, ∑ p ∈ (P i).parts, f ↑p + ↑ε'

Tactic:
by
      rw [Finset.sum_add_distrib]; norm_cast
      simp [show n * ε = ε' by field]

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

Goal state before:
pp: X : Type u_1
inst✝ : MeasurableSpace X
f : Set X → ℝ≥0∞
hf : IsSigmaSubadditiveSetFun f
hf' : f ∅ = 0
s : ℕ → Set X
hs : ∀ (i : ℕ), MeasurableSet (s i)
hs' : Pairwise (Disjoint on s)
b : ℝ≥0∞
hb : ∃ i, b < ∑ p ∈ i.parts, f ↑p
⊢ ∃ I, b < ∑ i ∈ I, preVariationFun f (s i)

Tactic:
obtain ⟨Q, hQ⟩ := hb

Goal state after:
X : Type u_1
inst✝ : MeasurableSpace X
f : Set X → ℝ≥0∞
hf : IsSigmaSubadditiveSetFun f
hf' : f ∅ = 0
s : ℕ → Set X
hs : ∀ (i : ℕ), MeasurableSet (s i)
hs' : Pairwise (Disjoint on s)
b : ℝ≥0∞
Q : Finpartition ⟨⋃ i, s i, ⋯⟩
hQ : b < ∑ p ∈ Q.parts, f ↑p
⊢ ∃ I, b < ∑ i ∈ I, preVariationFun f (s i)
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: case refine_2.inr
E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K) n)
A : ∀ (n : ℕ), IsCompact ((partialSups K) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} ∈ f
ν : ℕ → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n)).InnerRegular
νK : ∀ (n : ℕ), (ν n) ((partialSups K) n)ᶜ = 0
hν : ∀ (n : ℕ), Tendsto (fun ρ ↦ ρ.restrict (disjointed K n)) (↑f) (𝓝 (ν n))
B : (Measure.sum fun n ↦ ↑(ν n)) univ ≤ ↑C
μ : FiniteMeasure E := ⟨Measure.sum fun n ↦ ↑(ν n), ⋯⟩
L : Tendsto id (↑f) (𝓝 μ)
n : ℕ
h : Monotone K
⊢ ↑μ (K n)ᶜ ≤ ↑(u n)

Tactic:
simp only [toMeasure_mk, (hK n).measurableSet.compl, Measure.sum_apply, μ]

Goal state after:
case refine_2.inr
E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K) n)
A : ∀ (n : ℕ), IsCompact ((partialSups K) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} ∈ f
ν : ℕ → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n)).InnerRegular
νK : ∀ (n : ℕ), (ν n) ((partialSups K) n)ᶜ = 0
hν : ∀ (n : ℕ), Tendsto (fun ρ ↦ ρ.restrict (disjointed K n)) (↑f) (𝓝 (ν n))
B : (Measure.sum fun n ↦ ↑(ν n)) univ ≤ ↑C
μ : FiniteMeasure E := ⟨Measure.sum fun n ↦ ↑(ν n), ⋯⟩
L : Tendsto id (↑f) (𝓝 μ)
n : ℕ
h : Monotone K
⊢ ∑' (i : ℕ), ↑(ν i) (K n)ᶜ ≤ ↑(u n)
Module: Mathlib.MeasureTheory.Measure.ResolventTransform

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : MeasurableSpace 𝕜
inst✝⁹ : CompleteSpace 𝕜
inst✝⁸ : NormedDivisionRing A
inst✝⁷ : NormedAlgebra 𝕜 A
inst✝⁶ : HereditarilyLindelofSpace 𝕜
inst✝⁵ : OpensMeasurableSpace 𝕜
inst✝⁴ : CompleteSpace A
inst✝³ : SecondCountableTopology A
inst✝² : MeasurableSpace A
inst✝¹ : BorelSpace A
μ : Measure 𝕜
inst✝ : IsFiniteMeasure μ
a : A
hz : a ∉ ⇑(algebraMap 𝕜 A) '' μ.support
⊢ HasFiniteIntegral (resolvent a) μ

Tactic:
apply HasFiniteIntegral.of_bounded

Goal state after:
case hC
𝕜 : Type u_1
A : Type u_2
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : MeasurableSpace 𝕜
inst✝⁹ : CompleteSpace 𝕜
inst✝⁸ : NormedDivisionRing A
inst✝⁷ : NormedAlgebra 𝕜 A
inst✝⁶ : HereditarilyLindelofSpace 𝕜
inst✝⁵ : OpensMeasurableSpace 𝕜
inst✝⁴ : CompleteSpace A
inst✝³ : SecondCountableTopology A
inst✝² : MeasurableSpace A
inst✝¹ : BorelSpace A
μ : Measure 𝕜
inst✝ : IsFiniteMeasure μ
a : A
hz : a ∉ ⇑(algebraMap 𝕜 A) '' μ.support
⊢ ∀ᵐ (a_1 : 𝕜) ∂μ, ‖resolvent a a_1‖ ≤ ?C

case C
𝕜 : Type u_1
A : Type u_2
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : MeasurableSpace 𝕜
inst✝⁹ : CompleteSpace 𝕜
inst✝⁸ : NormedDivisionRing A
inst✝⁷ : NormedAlgebra 𝕜 A
inst✝⁶ : HereditarilyLindelofSpace 𝕜
inst✝⁵ : OpensMeasurableSpace 𝕜
inst✝⁴ : CompleteSpace A
inst✝³ : SecondCountableTopology A
inst✝² : MeasurableSpace A
inst✝¹ : BorelSpace A
μ : Measure 𝕜
inst✝ : IsFiniteMeasure μ
a : A
hz : a ∉ ⇑(algebraMap 𝕜 A) '' μ.support
⊢ ℝ
Module: Mathlib.MeasureTheory.Measure.SeparableMeasure

Goal state before:
pp: case inr
X : Type u_1
E : Type u_2
m : MeasurableSpace X
inst✝ : NormedAddCommGroup E
μ : Measure X
p : ℝ≥0∞
one_le_p : Fact (1 ≤ p)
p_ne_top : Fact (p ≠ ∞)
𝒜 : Set (Set X)
h𝒜 : μ.MeasureDense 𝒜
c : E
hc : c ≠ 0
p_pos : 0 < p
s : Set X
ms : MeasurableSet s
hμs : μ s ≠ ∞
ε : ℝ
hε : ε > 0
⊢ ∃ b ∈ {x | ∃ s, ∃ (hs : s ∈ 𝒜) (hμs : μ s ≠ ∞), indicatorConstLp p ⋯ hμs c = x},
    dist (indicatorConstLp p ms hμs c) b < ε

Tactic:
have aux : 0 < (ε / ‖c‖) ^ p.toReal := rpow_pos_of_pos (div_pos hε (norm_pos_iff.2 hc)) _

Goal state after:
case inr
X : Type u_1
E : Type u_2
m : MeasurableSpace X
inst✝ : NormedAddCommGroup E
μ : Measure X
p : ℝ≥0∞
one_le_p : Fact (1 ≤ p)
p_ne_top : Fact (p ≠ ∞)
𝒜 : Set (Set X)
h𝒜 : μ.MeasureDense 𝒜
c : E
hc : c ≠ 0
p_pos : 0 < p
s : Set X
ms : MeasurableSet s
hμs : μ s ≠ ∞
ε : ℝ
hε : ε > 0
aux : 0 < (ε / ‖c‖) ^ p.toReal
⊢ ∃ b ∈ {x | ∃ s, ∃ (hs : s ∈ 𝒜) (hμs : μ s ≠ ∞), indicatorConstLp p ⋯ hμs c = x},
    dist (indicatorConstLp p ms hμs c) b < ε
Module: Mathlib.MeasureTheory.Measure.ResolventTransform

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : HereditarilyLindelofSpace 𝕜
inst✝⁵ : CompleteSpace 𝕜
inst✝⁴ : MeasurableSpace 𝕜
inst✝³ : BorelSpace 𝕜
inst✝² : RCLike A
inst✝¹ : NormedAlgebra 𝕜 A
μ : Measure 𝕜
inst✝ : IsFiniteMeasure μ
a : A
ha : a ∉ ⇑(algebraMap 𝕜 A) '' μ.support
h : μ.support.Nonempty
this : 0 < infDist a (⇑(algebraMap 𝕜 A) '' μ.support)
s : Set A := ball a (infDist a (⇑(algebraMap 𝕜 A) '' μ.support) / 2)
hs_z : s ∈ 𝓝 a
⊢ ball a ?m.199 ⊆ (⇑(algebraMap 𝕜 A) '' μ.support)ᶜ

pp: case h
𝕜 : Type u_1
A : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : HereditarilyLindelofSpace 𝕜
inst✝⁵ : CompleteSpace 𝕜
inst✝⁴ : MeasurableSpace 𝕜
inst✝³ : BorelSpace 𝕜
inst✝² : RCLike A
inst✝¹ : NormedAlgebra 𝕜 A
μ : Measure 𝕜
inst✝ : IsFiniteMeasure μ
a : A
ha : a ∉ ⇑(algebraMap 𝕜 A) '' μ.support
h : μ.support.Nonempty
this : 0 < infDist a (⇑(algebraMap 𝕜 A) '' μ.support)
s : Set A := ball a (infDist a (⇑(algebraMap 𝕜 A) '' μ.support) / 2)
hs_z : s ∈ 𝓝 a
⊢ infDist a (⇑(algebraMap 𝕜 A) '' μ.support) / 2 ≤ ?m.199

pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : HereditarilyLindelofSpace 𝕜
inst✝⁵ : CompleteSpace 𝕜
inst✝⁴ : MeasurableSpace 𝕜
inst✝³ : BorelSpace 𝕜
inst✝² : RCLike A
inst✝¹ : NormedAlgebra 𝕜 A
μ : Measure 𝕜
inst✝ : IsFiniteMeasure μ
a : A
ha : a ∉ ⇑(algebraMap 𝕜 A) '' μ.support
h : μ.support.Nonempty
this : 0 < infDist a (⇑(algebraMap 𝕜 A) '' μ.support)
s : Set A := ⋯
hs_z : s ∈ 𝓝 a
⊢ ℝ

Tactic:
ball_infDist_subset_compl

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : HereditarilyLindelofSpace 𝕜
inst✝⁵ : CompleteSpace 𝕜
inst✝⁴ : MeasurableSpace 𝕜
inst✝³ : BorelSpace 𝕜
inst✝² : RCLike A
inst✝¹ : NormedAlgebra 𝕜 A
μ : Measure 𝕜
inst✝ : IsFiniteMeasure μ
a : A
ha : a ∉ ⇑(algebraMap 𝕜 A) '' μ.support
h : μ.support.Nonempty
this : 0 < infDist a (⇑(algebraMap 𝕜 A) '' μ.support)
s : Set A := ball a (infDist a (⇑(algebraMap 𝕜 A) '' μ.support) / 2)
hs_z : s ∈ 𝓝 a
⊢ ?m.221ᶜ ⊆ (⇑(algebraMap 𝕜 A) '' μ.support)ᶜ

𝕜 : Type u_1
A : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : HereditarilyLindelofSpace 𝕜
inst✝⁵ : CompleteSpace 𝕜
inst✝⁴ : MeasurableSpace 𝕜
inst✝³ : BorelSpace 𝕜
inst✝² : RCLike A
inst✝¹ : NormedAlgebra 𝕜 A
μ : Measure 𝕜
inst✝ : IsFiniteMeasure μ
a : A
ha : a ∉ ⇑(algebraMap 𝕜 A) '' μ.support
h : μ.support.Nonempty
this : 0 < infDist a (⇑(algebraMap 𝕜 A) '' μ.support)
s : Set A := ⋯
hs_z : s ∈ 𝓝 a
⊢ Set A

case h
𝕜 : Type u_1
A : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : HereditarilyLindelofSpace 𝕜
inst✝⁵ : CompleteSpace 𝕜
inst✝⁴ : MeasurableSpace 𝕜
inst✝³ : BorelSpace 𝕜
inst✝² : RCLike A
inst✝¹ : NormedAlgebra 𝕜 A
μ : Measure 𝕜
inst✝ : IsFiniteMeasure μ
a : A
ha : a ∉ ⇑(algebraMap 𝕜 A) '' μ.support
h : μ.support.Nonempty
this : 0 < infDist a (⇑(algebraMap 𝕜 A) '' μ.support)
s : Set A := ball a (infDist a (⇑(algebraMap 𝕜 A) '' μ.support) / 2)
hs_z : s ∈ 𝓝 a
⊢ infDist a (⇑(algebraMap 𝕜 A) '' μ.support) / 2 ≤ infDist a ?m.221
Module: Mathlib.MeasureTheory.Measure.WithDensityFinite

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsProbabilityMeasure μ
⊢ μ.toFinite = μ

Tactic:
rw [Measure.toFinite, Measure.toFiniteAux, if_pos, ProbabilityTheory.cond_univ]

Goal state after:
case hc
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsProbabilityMeasure μ
⊢ IsFiniteMeasure μ
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
μ : FiniteMeasure E
⊢ (∃ x, (∀ (n : ℕ), x (K n)ᶜ ≤ u n) ∧ x.toFiniteMeasure = μ) ↔ μ.mass = 1 ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n

Tactic:
refine ⟨?_, ?_⟩

Goal state after:
case refine_1
E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
μ : FiniteMeasure E
⊢ (∃ x, (∀ (n : ℕ), x (K n)ᶜ ≤ u n) ∧ x.toFiniteMeasure = μ) → μ.mass = 1 ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n

case refine_2
E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
μ : FiniteMeasure E
⊢ (μ.mass = 1 ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n) → ∃ x, (∀ (n : ℕ), x (K n)ᶜ ≤ u n) ∧ x.toFiniteMeasure = μ
Module: Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne

Goal state before:
pp: case neg
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 := fun n ↦ if h : μ (A n) = 1 then A n else (A n)ᶜ
mBn : MeasurableSet (⋂ n, B n)
n : ℕ
h : ¬μ (A n) = 1
⊢ μ (A n)ᶜᶜ = 0

Tactic:
rw [compl_compl]

Goal state after:
case neg
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 := fun n ↦ if h : μ (A n) = 1 then A n else (A n)ᶜ
mBn : MeasurableSet (⋂ n, B n)
n : ℕ
h : ¬μ (A n) = 1
⊢ μ (A n) = 0
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: 𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst✝² : PseudoMetricSpace 𝓧
inst✝¹ : OpensMeasurableSpace 𝓧
inst✝ : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
U : ℕ → Set 𝓧
O : ∀ (i : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U i)
Cov : ⋃ i, U i = univ
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
ε : ℝ
hε' : 0 ≤ ε
hε : 0 < ε
hεbound : ε ≤ 1
μ : ℕ → ProbabilityMeasure 𝓧
hμInS : ∀ (k : ℕ), μ k ∈ S
hcontradiction : ∀ (k : ℕ), ↑((μ k) (⋃ i, ⋃ (_ : i ≤ k), U i)) ≤ 1 - ↑(NNReal.mk ε hε')
μlim : ProbabilityMeasure 𝓧
left✝ : μlim ∈ closure[ProbabilityMeasure.instTopologicalSpace] S
sub : ℕ → ℕ
hsubmono : StrictMono sub
hμconverges : Tendsto (μ ∘ sub) atTop (𝓝 μlim)
n : ℕ
hopen : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (⋃ i, ⋃ (_ : i ≤ n), U i)
this : ↑μlim (⋃ i, ⋃ (_ : i ≤ n), U i) ≤ liminf (fun i ↦ ↑((μ ∘ sub) i) (⋃ i, ⋃ (_ : i ≤ n), U i)) atTop
⊢ ↑(μlim (⋃ i, ⋃ (_ : i ≤ n), U i)) ≤ liminf (fun k ↦ ↑((μ (sub k)) (⋃ i, ⋃ (_ : i ≤ n), U i))) atTop

Tactic:
Function.comp_apply,

Goal state after:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst✝² : PseudoMetricSpace 𝓧
inst✝¹ : OpensMeasurableSpace 𝓧
inst✝ : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
U : ℕ → Set 𝓧
O : ∀ (i : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U i)
Cov : ⋃ i, U i = univ
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
ε : ℝ
hε' : 0 ≤ ε
hε : 0 < ε
hεbound : ε ≤ 1
μ : ℕ → ProbabilityMeasure 𝓧
hμInS : ∀ (k : ℕ), μ k ∈ S
hcontradiction : ∀ (k : ℕ), ↑((μ k) (⋃ i, ⋃ (_ : i ≤ k), U i)) ≤ 1 - ↑(NNReal.mk ε hε')
μlim : ProbabilityMeasure 𝓧
left✝ : μlim ∈ closure[ProbabilityMeasure.instTopologicalSpace] S
sub : ℕ → ℕ
hsubmono : StrictMono sub
hμconverges : Tendsto (μ ∘ sub) atTop (𝓝 μlim)
n : ℕ
hopen : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (⋃ i, ⋃ (_ : i ≤ n), U i)
this : ↑μlim (⋃ i, ⋃ (_ : i ≤ n), U i) ≤ liminf (fun i ↦ ↑(μ (sub i)) (⋃ i, ⋃ (_ : i ≤ n), U i)) atTop
⊢ ↑(μlim (⋃ i, ⋃ (_ : i ≤ n), U i)) ≤ liminf (fun k ↦ ↑((μ (sub k)) (⋃ i, ⋃ (_ : i ≤ n), U i))) atTop
Module: Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne

Goal state before:
pp: 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 := fun n ↦ if h : μ (A n) = 1 then A n else (A n)ᶜ
mBn : MeasurableSet (⋂ n, B n)
hBn : μ (⋂ n, B n) = 1
⊢ ∃ x₀, μ = Measure.dirac x₀

Tactic:
obtain ⟨x₀, hx₀⟩ : ∃ x₀, ⋂ n, B n = {x₀} := by
    simp_rw [eq_singleton_iff_unique_mem]
    have neBn : (⋂ n, B n).Nonempty := by
      by_contra! h
      rw [h] at hBn
      simp_all
    refine ⟨neBn.some, neBn.some_mem, fun y hy ↦ ?_⟩
    refine hAsep y (by trivial) neBn.some (by trivial) fun n ↦ ?_
    have hsome := neBn.some_mem
    simp only [dite_eq_ite, mem_iInter, B] at hsome hy
    specialize hsome n
    specialize hy n
    constructor
    · intro h
      split_ifs at hy with hμAn
      · simpa [hμAn] using! hsome
      · contradiction
    · intro h
      split_ifs at hsome with hμAn
      · simpa [hμAn] using! hy
      · contradiction

Goal state after:
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 := fun n ↦ if h : μ (A n) = 1 then A n else (A n)ᶜ
mBn : MeasurableSet (⋂ n, B n)
hBn : μ (⋂ n, B n) = 1
x₀ : α
hx₀ : ⋂ n, B n = {x₀}
⊢ ∃ x₀, μ = Measure.dirac x₀
Module: Mathlib.MeasureTheory.OuterMeasure.OfAddContent

Goal state before:
pp: case refine_1
α : Type u_1
C : Set (Set α)
s : Set α
hC : IsSetSemiring C
m : AddContent ℝ≥0∞ C
m_sigma_subadd : m.IsSigmaSubadditive
hs : s ∈ C
⊢ (inducedOuterMeasure (fun x x_1 ↦ m x) ⋯ ⋯) s = (OuterMeasure.ofFunction ⇑(AddContent.extend hC m) ⋯) s

Tactic:
congr

Goal state after: