mathlib-initiative/mathlib-tactics huggingface dataset:
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
hs_μ : s ⊆ (⇑(algebraMap 𝕜 A) '' μ.support)ᶜ
resolvent_meas : ∀ᶠ (w : A) in 𝓝 a, AEStronglyMeasurable (resolvent w) μ
x : 𝕜
hx : x ∈ μ.support
w : A
hw : w ∈ s
⊢ ‖(algebraMap 𝕜 A) x - a‖ - ‖a - w‖ ≤ ‖(algebraMap 𝕜 A) x - w‖

Tactic:
by

Goal state after:
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
hs_μ : s ⊆ (⇑(algebraMap 𝕜 A) '' μ.support)ᶜ
resolvent_meas : ∀ᶠ (w : A) in 𝓝 a, AEStronglyMeasurable (resolvent w) μ
⊢ ∀ᵐ (x : 𝕜) ∂μ, ∀ w ∈ s, ‖resolvent w x ^ 2‖ ≤ (infDist a (⇑(algebraMap 𝕜 A) '' μ.support) / 2)⁻¹ ^ 2

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
E : Type u_3
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : PolishSpace E
inst✝ : BorelSpace E
ι : Type u_4
𝓕 : Filter ι
μ : ι → ProbabilityMeasure E
h_tight : IsTightMeasureSet {x | ∃ n, ↑(μ n) = x}
μ₀ : ProbabilityMeasure E
A : StarSubalgebra 𝕜 (E →ᵇ 𝕜)
hA : (StarSubalgebra.map (toContinuousMapStarₐ 𝕜) A).SeparatesPoints
hμ : ∀ g ∈ A, Tendsto (fun n ↦ ∫ (x : E), g x ∂↑(μ n)) 𝓕 (𝓝 (∫ (x : E), g x ∂↑μ₀))
⊢ Tendsto μ 𝓕 (𝓝 μ₀)

Tactic:
by

Goal state after:
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
hs_μ : s ⊆ (⇑(algebraMap 𝕜 A) '' μ.support)ᶜ
resolvent_meas : ∀ᶠ (w : A) in 𝓝 a, AEStronglyMeasurable (resolvent w) μ
resolvent'_bound : ∀ᵐ (x : 𝕜) ∂μ, ∀ w ∈ s, ‖resolvent w x ^ 2‖ ≤ (infDist a (⇑(algebraMap 𝕜 A) '' μ.support) / 2)⁻¹ ^ 2
⊢ ∀ᵐ (x : 𝕜) ∂μ, ∀ w ∈ s, HasDerivAt (fun w ↦ resolvent w x) (resolvent w x ^ 2) w

Tactic:
by

Goal state after:
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
hs_μ : s ⊆ (⇑(algebraMap 𝕜 A) '' μ.support)ᶜ
resolvent_meas : ∀ᶠ (w : A) in 𝓝 a, AEStronglyMeasurable (resolvent w) μ
resolvent'_bound : ∀ᵐ (x : 𝕜) ∂μ, ∀ w ∈ s, ‖resolvent w x ^ 2‖ ≤ (infDist a (⇑(algebraMap 𝕜 A) '' μ.support) / 2)⁻¹ ^ 2
h_deriv : ∀ᵐ (x : 𝕜) ∂μ, ∀ w ∈ s, HasDerivAt (fun w ↦ resolvent w x) (resolvent w x ^ 2) w
⊢ a ∉ ⇑(algebraMap 𝕜 A) '' μ.support

Tactic:
by

Goal state after:
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
hs_μ : s ⊆ (⇑(algebraMap 𝕜 A) '' μ.support)ᶜ
resolvent_meas : ∀ᶠ (w : A) in 𝓝 a, AEStronglyMeasurable (resolvent w) μ
resolvent'_bound : ∀ᵐ (x : 𝕜) ∂μ, ∀ w ∈ s, ‖resolvent w x ^ 2‖ ≤ (infDist a (⇑(algebraMap 𝕜 A) '' μ.support) / 2)⁻¹ ^ 2
h_deriv : ∀ᵐ (x : 𝕜) ∂μ, ∀ w ∈ s, HasDerivAt (fun w ↦ resolvent w x) (resolvent w x ^ 2) w
⊢ AEStronglyMeasurable (fun a_1 ↦ resolvent a a_1 ^ 2) μ

Tactic:
by

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

Goal state before:
pp: 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
⊢ ∫ (x : E), g x ∂μ = w.coeff.sum fun a z ↦ z * ∫ (x : E), ↑(probChar (((innerₗ E) x) a)) ∂μ

Tactic:
by

Goal state after:
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
hs_μ : s ⊆ (⇑(algebraMap 𝕜 A) '' μ.support)ᶜ
resolvent_meas : ∀ᶠ (w : A) in 𝓝 a, AEStronglyMeasurable (resolvent w) μ
resolvent'_bound : ∀ᵐ (x : 𝕜) ∂μ, ∀ w ∈ s, ‖resolvent w x ^ 2‖ ≤ (infDist a (⇑(algebraMap 𝕜 A) '' μ.support) / 2)⁻¹ ^ 2
h_deriv : ∀ᵐ (x : 𝕜) ∂μ, ∀ w ∈ s, HasDerivAt (fun w ↦ resolvent w x) (resolvent w x ^ 2) w
⊢ Integrable (fun a_1 ↦ (infDist a (⇑(algebraMap 𝕜 A) '' μ.support) / 2)⁻¹ ^ 2) μ

Tactic:
by

Goal state after:
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
⊢ HasDerivAt (resolventTransform μ) (∫ (x : 𝕜), resolvent a x ^ 2 ∂μ) a

Tactic:
by

Goal state after:
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
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
this✝¹ : Tendsto (fun m ↦ ∑ i ∈ Finset.range (m + 1), ↑(ν i) (K n)ᶜ) atTop (𝓝 (∑' (i : ℕ), ↑(ν i) (K n)ᶜ))
m : ℕ
hm : n ≤ m
this✝ : ∑ i ∈ Finset.range (m + 1), ↑(ν i) (K n)ᶜ = ∑ i ∈ Finset.Ioc n m, ↑(ν i) (K n)ᶜ
this :
  Tendsto (fun μ ↦ (∑ i ∈ Finset.Ioc n m, μ.restrict (disjointed K i)).mass) (↑f) (𝓝 (∑ i ∈ Finset.Ioc n m, ν i).mass)
μ : FiniteMeasure E
hμ : μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n
⊢ ∑ i ∈ Finset.Ioc n m, μ.restrict (disjointed K i) = μ.restrict (K m \ K n)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : HereditarilyLindelofSpace 𝕜
inst✝⁴ : CompleteSpace 𝕜
inst✝³ : MeasurableSpace 𝕜
inst✝² : BorelSpace 𝕜
inst✝¹ : NormedAlgebra 𝕜 ℂ
μ : Measure 𝕜
inst✝ : IsFiniteMeasure μ
⊢ AnalyticOn ℂ (resolventTransform μ) (⇑(algebraMap 𝕜 ℂ) '' μ.support)ᶜ

Tactic:
by

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

Goal state before:
pp: 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
hg : g ∈ charPoly continuous_probChar ⋯
⊢ Tendsto (fun n ↦ ∫ (x : E), g x ∂↑(μ n)) 𝓕 (𝓝 (∫ (x : E), g x ∂↑μ₀))

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
μ₀ : ProbabilityMeasure E
μ : ℕ → ProbabilityMeasure E
h : ∀ (t : E), Tendsto (fun n ↦ charFun (↑(μ n)) t) atTop (𝓝 (charFun (↑μ₀) t))
⊢ ContinuousAt (charFun ↑μ₀) 0

Tactic:
by

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν ξ : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : μ - ν ≤ ξ
s : Set α
hs : IsHahnDecomposition μ ν s
h_le_s : μ.restrict s ≤ ξ.restrict s + ν.restrict s
⊢ μ.restrict sᶜ ≤ ξ.restrict sᶜ + ν.restrict sᶜ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν ξ : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : μ - ν ≤ ξ
s : Set α
hs : IsHahnDecomposition μ ν s
h_le_s : μ.restrict s ≤ ξ.restrict s + ν.restrict s
h_le_s_compl : μ.restrict sᶜ ≤ ξ.restrict sᶜ + ν.restrict sᶜ
this : μ.restrict s + μ.restrict sᶜ ≤ ξ.restrict s + ν.restrict s + (ξ.restrict sᶜ + ν.restrict sᶜ)
⊢ ξ.restrict s + ν.restrict s + (ξ.restrict sᶜ + ν.restrict sᶜ) =
    ξ.restrict s + ξ.restrict sᶜ + (ν.restrict s + ν.restrict sᶜ)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν ξ : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
⊢ μ - ν ≤ ξ ↔ μ ≤ ξ + ν

Tactic:
by

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

Goal state before:
pp: case h₁
X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
inst✝ : IsFiniteMeasure μ
h𝒜 : IsSetAlgebra 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
f : ℕ → Set X
hs✝ : ∀ (n : ℕ), MeasurableSet (f n)
hf : ∀ (n : ℕ), MeasurableSet (f n) ∧ ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, μ.real (f n ∆ t) < ε
ε : ℝ
ε_pos : 0 < ε
this : Filter.Tendsto (fun n ↦ (μ (accumulate f n)).toReal) Filter.atTop (nhds (μ (⋃ i, f i)).toReal)
N : ℕ
hN : ∀ n ≥ N, dist (μ (accumulate f n)).toReal (μ (⋃ i, f i)).toReal < ε / 2
g : ℕ → Set X
g_mem : ∀ (n : ℕ), g n ∈ 𝒜
hg : ∀ (n : ℕ), μ.real (f n ∆ g n) < ε / (2 * (↑N + 1))
⊢ (μ (⋃ n, f n)).toReal - (μ (⋃ n ∈ Finset.range (N + 1), f n)).toReal < ε / 2

Tactic:
apply lt_of_le_of_lt (sub_le_dist ..)
                    simp only [Finset.mem_range, Nat.lt_add_one_iff]
                    exact (dist_comm (α := ℝ) .. ▸ hN N (le_refl N))

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

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
μ₀ : ProbabilityMeasure E
μ : ℕ → ProbabilityMeasure E
h : ∀ (t : E), Tendsto (fun n ↦ charFun (↑(μ n)) t) atTop (𝓝 (charFun (↑μ₀) t))
⊢ Tendsto μ atTop (𝓝 μ₀)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
inst✝ : IsFiniteMeasure (μ.withDensity g)
hg : Measurable g
hgf : g ≤ᵐ[μ] f
⊢ μ.withDensity (f - g) = μ.withDensity f - μ.withDensity g

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
μ₀ : ProbabilityMeasure E
μ : ℕ → ProbabilityMeasure E
⊢ Tendsto μ atTop (𝓝 μ₀) ↔ ∀ (t : E), Tendsto (fun n ↦ charFun (↑(μ n)) t) atTop (𝓝 (charFun (↑μ₀) t))

Tactic:
by

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

Goal state before:
pp: 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}
ht : MeasurableSet t
⊢ Measurable (f - g)

Tactic:
by

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

Goal state before:
pp: 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}
ht : MeasurableSet t
⊢ (μ.withDensity (f - g)).restrict t = 0

Tactic:
by

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

Goal state before:
pp: 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}
ht : MeasurableSet t
h_zero : (μ.withDensity (f - g)).restrict t = 0
⊢ IsFiniteMeasure ((μ.restrict tᶜ).withDensity g)

Tactic:
by

Goal state after:
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
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
⊢ IsCompact {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n}

Tactic:
by

Goal state after:
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
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
⊢ {μ | μ.mass = C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} = {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} ∩ {μ | μ.mass = C}

Tactic:
by

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

Goal state before:
pp: 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:
by

Goal state after:
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
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
this : {μ | μ.mass = C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} = {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} ∩ {μ | μ.mass = C}
⊢ Continuous[FiniteMeasure.instTopologicalSpace, _] mass

Tactic:
by

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
⊢ ae μ.toFiniteAux = ae μ

Tactic:
Measure.toFiniteAux

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
⊢ ae (if IsFiniteMeasure μ then μ else ⋯.choose) = ae μ
Module: Mathlib.MeasureTheory.Measure.WithDensityFinite

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
⊢ ae μ.toFiniteAux = ae μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
⊢ IsFiniteMeasure μ.toFiniteAux

Tactic:
Measure.toFiniteAux

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
⊢ IsFiniteMeasure (if IsFiniteMeasure μ then μ else ⋯.choose)
Module: Mathlib.MeasureTheory.Measure.WithDensityFinite

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
⊢ IsFiniteMeasure μ.toFiniteAux

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
s : Set α
⊢ μ.toFinite s = 0 ↔ μ s = 0

Tactic:
by

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
⊢ μ.toFinite = 0 ↔ μ = 0

Tactic:
by

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
⊢ Measure.toFinite 0 = 0

Tactic:
by

Goal state after:
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
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
this : {μ | μ.mass = C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} = {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} ∩ {μ | μ.mass = C}
⊢ Continuous[FiniteMeasure.instTopologicalSpace, _] fun μ ↦ C

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsZeroOneMeasure μ
⊢ ∀ (s : Set α), μ s = 0 ∨ μ s = 1

Tactic:
by

Goal state after:
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:
by

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

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

Tactic:
by

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝¹ : SFinite μ
inst✝ : NeZero μ
⊢ IsProbabilityMeasure μ.toFinite

Tactic:
by

Goal state after:
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
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
⊢ IsCompact {μ | μ.mass = C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n}

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsZeroOneMeasure μ
⊢ (∃ s, μ s = 1) ↔ μ univ = 1

Tactic:
by

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)
inst✝ : IsFiniteMeasure μ
h𝒜 : IsSetAlgebra 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
f : ℕ → Set X
hs✝ : ∀ (n : ℕ), MeasurableSet (f n)
hf : ∀ (n : ℕ), MeasurableSet (f n) ∧ ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, μ.real (f n ∆ t) < ε
ε : ℝ
ε_pos : 0 < ε
this : Filter.Tendsto (fun n ↦ (μ (accumulate f n)).toReal) Filter.atTop (nhds (μ (⋃ i, f i)).toReal)
N : ℕ
hN : ∀ n ≥ N, dist (μ (accumulate f n)).toReal (μ (⋃ i, f i)).toReal < ε / 2
g : ℕ → Set X
g_mem : ∀ (n : ℕ), g n ∈ 𝒜
hg : ∀ (n : ℕ), μ.real (f n ∆ g n) < ε / (2 * (↑N + 1))
⊢ μ.real (⋃ n ∈ Finset.range (N + 1), f n ∆ g n) ≤ ∑ n ∈ Finset.range (N + 1), μ.real (f n ∆ g n)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
⊢ μ.restrict μ.sigmaFiniteSetᶜ = ∞ • μ.toFinite.restrict μ.sigmaFiniteSetᶜ

Tactic:
by

Goal state after:
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
hμ : μ.mass = 1
h'μ : ∀ (n : ℕ), μ (K n)ᶜ ≤ u n
⊢ (↑μ).real univ = 1

Tactic:
by

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)
inst✝ : IsFiniteMeasure μ
h𝒜 : IsSetAlgebra 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
f : ℕ → Set X
hs✝ : ∀ (n : ℕ), MeasurableSet (f n)
hf : ∀ (n : ℕ), MeasurableSet (f n) ∧ ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, μ.real (f n ∆ t) < ε
ε : ℝ
ε_pos : 0 < ε
this : Filter.Tendsto (fun n ↦ (μ (accumulate f n)).toReal) Filter.atTop (nhds (μ (⋃ i, f i)).toReal)
N : ℕ
hN : ∀ n ≥ N, dist (μ (accumulate f n)).toReal (μ (⋃ i, f i)).toReal < ε / 2
g : ℕ → Set X
g_mem : ∀ (n : ℕ), g n ∈ 𝒜
hg : ∀ (n : ℕ), μ.real (f n ∆ g n) < ε / (2 * (↑N + 1))
⊢ 0 ∈ Finset.range (N + 1)

Tactic:
by

Goal state after:
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
hμ : μ.mass = 1
h'μ : ∀ (n : ℕ), μ (K n)ᶜ ≤ u n
ν : ProbabilityMeasure E := ⟨↑μ, ⋯⟩
⊢ ν.toFiniteMeasure = μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsZeroOneMeasure μ
s t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
hμs : μ s = 1
hμt : μ t = 1
h✝ : μ (s ∩ t) = 0
this : μ (s ∩ t)ᶜ ≤ 0
⊢ μ (s ∩ t) ≠ ⊤

Tactic:
by

Goal state after:
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
hμ : μ.mass = 1
h'μ : ∀ (n : ℕ), μ (K n)ᶜ ≤ u n
ν : ProbabilityMeasure E := ⟨↑μ, ⋯⟩
this : ν.toFiniteMeasure = μ
⊢ ∀ (n : ℕ), ν (K n)ᶜ ≤ u n

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsZeroOneMeasure μ
s t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
hμs : μ s = 1
hμt : μ t = 1
h✝ : μ (s ∩ t) = 0
this : μ (s ∩ t)ᶜ ≤ 0
⊢ False

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsZeroOneMeasure μ
s t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
hμs : μ s = 1
hμt : μ t = 1
h✝ : μ (s ∩ t) = 0
⊢ μ (s ∩ t)ᶜ = μ (sᶜ ∪ tᶜ)

Tactic:
by

Goal state after:
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
⊢ ProbabilityMeasure.toFiniteMeasure '' {μ | ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} = {μ | μ.mass = 1 ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n}

Tactic:
by

Goal state after: