feat: Lévy-Prokhorov distance (#8719)

This PR adds the Lévy-Prokhorov distance on the space of finite measures on an emetric space, and shows that it is a pseudoemetric.



Co-authored-by: kkytola <“[email protected]”>
Co-authored-by: Yury G. Kudryashov <[email protected]>
Co-authored-by: kkytola <[email protected]>

Author: 3 people

Mathlib.lean

@@ -2793,6 +2793,7 @@ import Mathlib.MeasureTheory.Measure.Lebesgue.Complex
 import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
 import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
 import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
+import Mathlib.MeasureTheory.Measure.LevyProkhorovMetric
 import Mathlib.MeasureTheory.Measure.LogLikelihoodRatio
 import Mathlib.MeasureTheory.Measure.MeasureSpace
 import Mathlib.MeasureTheory.Measure.MeasureSpaceDef

Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean

@@ -0,0 +1,224 @@
+/-
+Copyright (c) 2023 Kalle Kytölä. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Kalle Kytölä
+-/
+import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
+
+/-!
+# The Lévy-Prokhorov distance on spaces of finite measures and probability measures
+
+## Main definitions
+
+* `MeasureTheory.levyProkhorovEDist`: The Lévy-Prokhorov edistance between two measures.
+* `MeasureTheory.levyProkhorovDist`: The Lévy-Prokhorov distance between two finite measures.
+
+## Main results
+
+* `levyProkhorovDist_pseudoMetricSpace_finiteMeasure`: The Lévy-Prokhorov distance is a
+  pseudoemetric on the space of finite measures.
+* `levyProkhorovDist_pseudoMetricSpace_probabilityMeasure`: The Lévy-Prokhorov distance is a
+  pseudoemetric on the space of probability measures.
+
+## Todo
+
+* Show that in Borel spaces, the Lévy-Prokhorov distance is a metric; not just a pseudometric.
+* Prove that if `X` is metrizable and separable, then the Lévy-Prokhorov distance metrizes the
+  topology of weak convergence.
+
+## Tags
+
+finite measure, probability measure, weak convergence, convergence in distribution, metrizability
+-/
+
+open Topology Metric Filter Set ENNReal NNReal
+
+namespace MeasureTheory
+
+open scoped Topology ENNReal NNReal BoundedContinuousFunction
+
+section Levy_Prokhorov
+
+/-! ### Lévy-Prokhorov metric -/
+
+variable {ι : Type*} {Ω : Type*} [MeasurableSpace Ω] [MetricSpace Ω]
+
+/-- The Lévy-Prokhorov edistance between measures:
+`d(μ,ν) = inf {r ≥ 0 | ∀ B, μ B ≤ ν Bᵣ + r ∧ ν B ≤ μ Bᵣ + r}`. -/
+noncomputable def levyProkhorovEDist (μ ν : Measure Ω) : ℝ≥0∞ :=
+  sInf {ε | ∀ B, MeasurableSet B →
+            μ B ≤ ν (thickening ε.toReal B) + ε ∧ ν B ≤ μ (thickening ε.toReal B) + ε}
+
+/- This result is not placed in earlier more generic files, since it is rather specialized;
+it mixes measure and metric in a very particular way. -/
+lemma meas_le_of_le_of_forall_le_meas_thickening_add {ε₁ ε₂ : ℝ≥0∞} (μ ν : Measure Ω)
+    (h_le : ε₁ ≤ ε₂) {B : Set Ω} (hε₁ : μ B ≤ ν (thickening ε₁.toReal B) + ε₁) :
+    μ B ≤ ν (thickening ε₂.toReal B) + ε₂ := by
+  by_cases ε_top : ε₂ = ∞
+  · simp only [ne_eq, FiniteMeasure.ennreal_coeFn_eq_coeFn_toMeasure, ε_top, top_toReal,
+                add_top, le_top]
+  apply hε₁.trans (add_le_add ?_ h_le)
+  exact measure_mono (μ := ν) (thickening_mono (toReal_mono ε_top h_le) B)
+
+lemma left_measure_le_of_levyProkhorovEDist_lt {μ ν : Measure Ω} {c : ℝ≥0∞}
+    (h : levyProkhorovEDist μ ν < c) {B : Set Ω} (B_mble : MeasurableSet B) :
+    μ B ≤ ν (thickening c.toReal B) + c := by
+  obtain ⟨c', ⟨hc', lt_c⟩⟩ := sInf_lt_iff.mp h
+  exact meas_le_of_le_of_forall_le_meas_thickening_add μ ν lt_c.le (hc' B B_mble).1
+
+lemma right_measure_le_of_levyProkhorovEDist_lt {μ ν : Measure Ω} {c : ℝ≥0∞}
+    (h : levyProkhorovEDist μ ν < c) {B : Set Ω} (B_mble : MeasurableSet B) :
+    ν B ≤ μ (thickening c.toReal B) + c := by
+  obtain ⟨c', ⟨hc', lt_c⟩⟩ := sInf_lt_iff.mp h
+  exact meas_le_of_le_of_forall_le_meas_thickening_add ν μ lt_c.le (hc' B B_mble).2
+
+lemma levyProkhorovEDist_le_of_forall_add_pos_le (μ ν : Measure Ω) (δ : ℝ≥0∞)
+    (h : ∀ ε B, 0 < ε → ε < ∞ → MeasurableSet B →
+      μ B ≤ ν (thickening (δ + ε).toReal B) + δ + ε ∧
+      ν B ≤ μ (thickening (δ + ε).toReal B) + δ + ε) :
+    levyProkhorovEDist μ ν ≤ δ := by
+  apply ENNReal.le_of_forall_pos_le_add
+  intro ε hε _
+  by_cases ε_top : ε = ∞
+  · simp only [ε_top, add_top, le_top]
+  apply sInf_le
+  intro B B_mble
+  simpa only [add_assoc] using h ε B (coe_pos.mpr hε) coe_lt_top B_mble
+
+lemma levyProkhorovEDist_le_of_forall (μ ν : Measure Ω) (δ : ℝ≥0∞)
+    (h : ∀ ε B, δ < ε → ε < ∞ → MeasurableSet B →
+        μ B ≤ ν (thickening ε.toReal B) + ε ∧ ν B ≤ μ (thickening ε.toReal B) + ε) :
+    levyProkhorovEDist μ ν ≤ δ := by
+  by_cases δ_top : δ = ∞
+  · simp only [δ_top, add_top, le_top]
+  apply levyProkhorovEDist_le_of_forall_add_pos_le
+  intro x B x_pos x_lt_top B_mble
+  simpa only [← add_assoc] using h (δ + x) B (ENNReal.lt_add_right δ_top x_pos.ne.symm)
+    (by simp only [add_lt_top, Ne.lt_top δ_top, x_lt_top, and_self]) B_mble
+
+lemma levyProkhorovEDist_le_max_measure_univ (μ ν : Measure Ω) :
+    levyProkhorovEDist μ ν ≤ max (μ univ) (ν univ) := by
+  refine sInf_le fun B _ ↦ ⟨?_, ?_⟩ <;> apply le_add_left <;> simp [measure_mono]
+
+lemma levyProkhorovEDist_lt_top (μ ν : Measure Ω) [IsFiniteMeasure μ] [IsFiniteMeasure ν] :
+    levyProkhorovEDist μ ν < ∞ :=
+  (levyProkhorovEDist_le_max_measure_univ μ ν).trans_lt <| by simp [measure_lt_top]
+
+lemma levyProkhorovEDist_self (μ : Measure Ω) :
+    levyProkhorovEDist μ μ = 0 := by
+  rw [← nonpos_iff_eq_zero, ← csInf_Ioo zero_lt_top]
+  refine sInf_le_sInf fun ε ⟨hε₀, hε_top⟩ B _ ↦ and_self_iff.2 ?_
+  refine le_add_right <| measure_mono <| self_subset_thickening ?_ _
+  exact ENNReal.toReal_pos hε₀.ne' hε_top.ne
+
+lemma levyProkhorovEDist_comm (μ ν : Measure Ω) :
+    levyProkhorovEDist μ ν = levyProkhorovEDist ν μ := by
+  simp only [levyProkhorovEDist, and_comm]
+
+lemma levyProkhorovEDist_triangle [OpensMeasurableSpace Ω] (μ ν κ : Measure Ω) :
+    levyProkhorovEDist μ κ ≤ levyProkhorovEDist μ ν + levyProkhorovEDist ν κ := by
+  by_cases LPμν_finite : levyProkhorovEDist μ ν = ∞
+  · simp [LPμν_finite]
+  by_cases LPνκ_finite : levyProkhorovEDist ν κ = ∞
+  · simp [LPνκ_finite]
+  apply levyProkhorovEDist_le_of_forall_add_pos_le
+  intro ε B ε_pos ε_lt_top B_mble
+  have half_ε_pos : 0 < ε / 2 := ENNReal.div_pos ε_pos.ne' two_ne_top
+  have half_ε_lt_top : ε / 2 < ∞ := ENNReal.div_lt_top ε_lt_top.ne two_ne_zero
+  let r := levyProkhorovEDist μ ν + ε / 2
+  let s := levyProkhorovEDist ν κ + ε / 2
+  have lt_r : levyProkhorovEDist μ ν < r := lt_add_right LPμν_finite half_ε_pos.ne'
+  have lt_s : levyProkhorovEDist ν κ < s := lt_add_right LPνκ_finite half_ε_pos.ne'
+  have hs_add_r : s + r = levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε := by
+    simp_rw [add_assoc, add_comm (ε / 2), add_assoc, ENNReal.add_halves, ← add_assoc,
+      add_comm (levyProkhorovEDist μ ν)]
+  have hs_add_r' : s.toReal + r.toReal
+      = (levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε).toReal := by
+    rw [← hs_add_r, ← ENNReal.toReal_add]
+    · exact ENNReal.add_ne_top.mpr ⟨LPνκ_finite, half_ε_lt_top.ne⟩
+    · exact ENNReal.add_ne_top.mpr ⟨LPμν_finite, half_ε_lt_top.ne⟩
+  rw [← hs_add_r', add_assoc, ← hs_add_r, add_assoc _ _ ε, ← hs_add_r]
+  refine ⟨?_, ?_⟩
+  · calc μ B ≤ ν (thickening r.toReal B) + r :=
+      left_measure_le_of_levyProkhorovEDist_lt lt_r B_mble
+    _ ≤ κ (thickening s.toReal (thickening r.toReal B)) + s + r :=
+      add_le_add_right
+        (left_measure_le_of_levyProkhorovEDist_lt lt_s isOpen_thickening.measurableSet) _
+    _ = κ (thickening s.toReal (thickening r.toReal B)) + (s + r) := add_assoc _ _ _
+    _ ≤ κ (thickening (s.toReal + r.toReal) B) + (s + r) :=
+      add_le_add_right (measure_mono (thickening_thickening_subset _ _ _)) _
+  · calc κ B ≤ ν (thickening s.toReal B) + s :=
+      right_measure_le_of_levyProkhorovEDist_lt lt_s B_mble
+    _ ≤ μ (thickening r.toReal (thickening s.toReal B)) + r + s :=
+      add_le_add_right
+        (right_measure_le_of_levyProkhorovEDist_lt lt_r isOpen_thickening.measurableSet) s
+    _ = μ (thickening r.toReal (thickening s.toReal B)) + (s + r) := by rw [add_assoc, add_comm r]
+    _ ≤ μ (thickening (r.toReal + s.toReal) B) + (s + r) :=
+      add_le_add_right (measure_mono (thickening_thickening_subset _ _ _)) _
+    _ = μ (thickening (s.toReal + r.toReal) B) + (s + r) := by rw [add_comm r.toReal]
+
+/-- The Lévy-Prokhorov distance between finite measures:
+`d(μ,ν) = inf {r ≥ 0 | ∀ B, μ B ≤ ν Bᵣ + r ∧ ν B ≤ μ Bᵣ + r}`. -/
+noncomputable def levyProkhorovDist (μ ν : Measure Ω) : ℝ :=
+  (levyProkhorovEDist μ ν).toReal
+
+lemma levyProkhorovDist_self (μ : Measure Ω) :
+    levyProkhorovDist μ μ = 0 := by
+  simp only [levyProkhorovDist, levyProkhorovEDist_self, zero_toReal]
+
+lemma levyProkhorovDist_comm (μ ν : Measure Ω) :
+    levyProkhorovDist μ ν = levyProkhorovDist ν μ := by
+  simp only [levyProkhorovDist, levyProkhorovEDist_comm]
+
+lemma levyProkhorovDist_triangle [OpensMeasurableSpace Ω] (μ ν κ : Measure Ω)
+    [IsFiniteMeasure μ] [IsFiniteMeasure ν] [IsFiniteMeasure κ] :
+    levyProkhorovDist μ κ ≤ levyProkhorovDist μ ν + levyProkhorovDist ν κ := by
+  have dμκ_finite := (levyProkhorovEDist_lt_top μ κ).ne
+  have dμν_finite := (levyProkhorovEDist_lt_top μ ν).ne
+  have dνκ_finite := (levyProkhorovEDist_lt_top ν κ).ne
+  convert (ENNReal.toReal_le_toReal (a := levyProkhorovEDist μ κ)
+    (b := levyProkhorovEDist μ ν + levyProkhorovEDist ν κ)
+    _ _).mpr <| levyProkhorovEDist_triangle μ ν κ
+  · simp only [levyProkhorovDist, ENNReal.toReal_add dμν_finite dνκ_finite]
+  · exact dμκ_finite
+  · exact ENNReal.add_ne_top.mpr ⟨dμν_finite, dνκ_finite⟩
+
+/-- A type synonym, to be used for `Measure α`, `FiniteMeasure α`, or `ProbabilityMeasure α`,
+when they are to be equipped with the Lévy-Prokhorov distance. -/
+def LevyProkhorov (α : Type*) := α
+
+variable [OpensMeasurableSpace Ω]
+
+/-- The Lévy-Prokhorov distance `levyProkhorovEDist` makes `Measure Ω` a pseudoemetric
+space. The instance is recorded on the type synonym `LevyProkhorov (Measure Ω) := Measure Ω`. -/
+noncomputable instance : PseudoEMetricSpace (LevyProkhorov (Measure Ω)) where
+  edist := levyProkhorovEDist
+  edist_self := levyProkhorovEDist_self
+  edist_comm := levyProkhorovEDist_comm
+  edist_triangle := levyProkhorovEDist_triangle
+
+/-- The Lévy-Prokhorov distance `levyProkhorovDist` makes `FiniteMeasure Ω` a pseudometric
+space. The instance is recorded on the type synonym
+`LevyProkhorov (FiniteMeasure Ω) := FiniteMeasure Ω`. -/
+noncomputable instance levyProkhorovDist_pseudoMetricSpace_finiteMeasure :
+    PseudoMetricSpace (LevyProkhorov (FiniteMeasure Ω)) where
+  dist μ ν := levyProkhorovDist μ.toMeasure ν.toMeasure
+  dist_self μ := levyProkhorovDist_self _
+  dist_comm μ ν := levyProkhorovDist_comm _ _
+  dist_triangle μ ν κ := levyProkhorovDist_triangle _ _ _
+  edist_dist μ ν := by simp [← ENNReal.ofReal_coe_nnreal]
+
+/-- The Lévy-Prokhorov distance `levyProkhorovDist` makes `ProbabilityMeasure Ω` a pseudoemetric
+space. The instance is recorded on the type synonym
+`LevyProkhorov (ProbabilityMeasure Ω) := ProbabilityMeasure Ω`. -/
+noncomputable instance levyProkhorovDist_pseudoMetricSpace_probabilityMeasure :
+    PseudoMetricSpace (LevyProkhorov (ProbabilityMeasure Ω)) where
+  dist μ ν := levyProkhorovDist μ.toMeasure ν.toMeasure
+  dist_self μ := levyProkhorovDist_self _
+  dist_comm μ ν := levyProkhorovDist_comm _ _
+  dist_triangle μ ν κ := levyProkhorovDist_triangle _ _ _
+  edist_dist μ ν := by simp [← ENNReal.ofReal_coe_nnreal]
+
+end Levy_Prokhorov --section
+
+end MeasureTheory -- namespace