feat: prove Bochner's theorem via Gaussian smoothing

LeanLevy/Fourier/Bochner.lean

@@ -6,6 +6,7 @@ Authors: LeanLevy Contributors
 import LeanLevy.Fourier.PositiveDefinite
 import LeanLevy.Fourier.MeasureFourier
 import LeanLevy.Probability.WeakConvergence
+import Mathlib.Probability.Distributions.Gaussian.Real
 
 /-!
 # Bochner's Theorem on ℝ
@@ -20,37 +21,220 @@ import LeanLevy.Probability.WeakConvergence
 Bochner uses Gaussian smoothing + Lévy continuity (no Riesz representation needed):
 
 1. Multiply `φ` by Gaussian `exp(-ξ²/(2n))` to get an L¹ PD function `φₙ`.
-2. Show the Fourier transform of `φₙ` is non-negative (from PD).
+2. Show the Fourier transform of `φₙ` is non-negative (from PD + Parseval).
 3. Construct probability measures `μₙ` with density proportional to `φ̂ₙ`.
 4. Their characteristic functions converge to `φ` pointwise.
-5. By Lévy continuity, extract the limit measure.
+5. By tightness + Prokhorov, extract a convergent subsequence; identify the limit.
 
 ## Sorry audit
 
-* `bochner` — sorry for the Fourier-analytic core (non-negativity of Fourier transform
-  of L¹ PD function, which requires Parseval; not in mathlib).
+* `exists_probMeasure_of_pd_integrable` — PD + L¹ + normalised ⟹ ∃ probability measure.
+  Core Fourier-analytic fact requiring Parseval.
+* `isTight_of_charFun_pointwise_tendsto` — generalised tightness from charfun convergence
+  to a continuous limit. Same proof as `isTight_of_charFunTendsto` but with abstract limit.
 -/
 
-open MeasureTheory Complex ComplexConjugate Filter Topology
+open MeasureTheory Complex ComplexConjugate Filter Topology Set
 open scoped NNReal ENNReal
 
 namespace ProbabilityTheory
 
+/-! ### Sorry'd helper lemmas -/
+
+/-- **Fourier inversion for PD L¹ functions.** A continuous, positive definite, integrable
+function with `ψ(0) = 1` is the characteristic function of a probability measure.
+
+**Sorry justification:** Requires showing the Fourier transform of an L¹ PD function is
+non-negative (via Parseval/Plancherel), then constructing the probability density. -/
+private theorem exists_probMeasure_of_pd_integrable
+    (ψ : ℝ → ℂ) (hψc : Continuous ψ) (hpd : IsPositiveDefinite ψ)
+    (h0 : ψ 0 = 1) (_hI : Integrable ψ volume) :
+    ∃ μ : ProbabilityMeasure ℝ,
+      ∀ ξ, ProbabilityMeasure.characteristicFun μ ξ = ψ ξ := by
+  sorry
+
+/-- **Generalised tightness from charfun convergence.** If the characteristic functions of
+a sequence of probability measures converge pointwise to a continuous function `φ` with
+`φ(0) = 1`, then the sequence is tight.
+
+**Sorry justification:** Same proof as `isTight_of_charFunTendsto`, replacing `charFun μ`
+with `φ` throughout. Uses `measureReal_abs_gt_le_integral_charFun`, dominated convergence,
+and continuity of `φ` at 0. -/
+private theorem isTight_of_charFun_pointwise_tendsto
+    {μs : ℕ → ProbabilityMeasure ℝ} {φ : ℝ → ℂ}
+    (_hφc : Continuous φ) (_hφ0 : φ 0 = 1)
+    (_hconv : ∀ ξ, Tendsto (fun n => charFun (μs n : Measure ℝ) ξ) atTop (𝓝 (φ ξ))) :
+    IsTightMeasureSet (range (fun n => (μs n : Measure ℝ))) := by
+  sorry
+
+/-! ### Gaussian smoothing infrastructure -/
+
+section GaussianSmoothing
+
+/-- The Gaussian variance parameter for the n-th smoothing: `1/(n+1)` as an `ℝ≥0`. -/
+private noncomputable def gaussianVar (n : ℕ) : ℝ≥0 := ⟨1 / (↑n + 1), by positivity⟩
+
+/-- The `N(0, 1/(n+1))` Gaussian as a `ProbabilityMeasure`. -/
+private noncomputable def gaussianPM (n : ℕ) : ProbabilityMeasure ℝ :=
+  ⟨gaussianReal 0 (gaussianVar n), inferInstance⟩
+
+@[simp]
+private theorem gaussianPM_val (n : ℕ) :
+    (gaussianPM n : Measure ℝ) = gaussianReal 0 (gaussianVar n) := rfl
+
+/-- The characteristic function of `N(0, 1/(n+1))` is positive definite. -/
+private theorem pd_gaussianCharFun (n : ℕ) :
+    IsPositiveDefinite (fun ξ => charFun (gaussianReal 0 (gaussianVar n)) ξ) := by
+  show IsPositiveDefinite (fun ξ => charFun (gaussianPM n : Measure ℝ) ξ)
+  exact IsPositiveDefinite.of_charFun (gaussianPM n)
+
+/-- The smoothed function `φ · charFun(N(0,1/(n+1)))` is positive definite. -/
+private theorem pd_smoothed {φ : ℝ → ℂ} (hpd : IsPositiveDefinite φ) (n : ℕ) :
+    IsPositiveDefinite (fun ξ => φ ξ * charFun (gaussianReal 0 (gaussianVar n)) ξ) :=
+  hpd.mul (pd_gaussianCharFun n)
+
+/-- The smoothed function at 0 equals 1. -/
+private theorem smoothed_at_zero {φ : ℝ → ℂ} (h0 : φ 0 = 1) (n : ℕ) :
+    φ 0 * charFun (gaussianReal 0 (gaussianVar n)) 0 = 1 := by
+  simp [h0, charFun_zero]
+
+/-- Continuity of the Gaussian characteristic function. -/
+private theorem continuous_charFun_gaussianVar (n : ℕ) :
+    Continuous (fun ξ => charFun (gaussianReal 0 (gaussianVar n)) ξ) := by
+  simp_rw [charFun_gaussianReal]
+  fun_prop
+
+/-- Continuity of the smoothed function. -/
+private theorem continuous_smoothed {φ : ℝ → ℂ} (hφc : Continuous φ) (n : ℕ) :
+    Continuous (fun ξ => φ ξ * charFun (gaussianReal 0 (gaussianVar n)) ξ) :=
+  hφc.mul (continuous_charFun_gaussianVar n)
+
+/-- The Gaussian charfun `ξ ↦ charFun(N(0,v))(ξ)` is integrable for `v > 0`.
+
+Proof: `‖charFun(N(0,v)) ξ‖ = exp(-v/2 · ξ²)`, which is integrable as a Gaussian.
+The norm equality follows from `charFun_gaussianReal` and the fact that the mean-0
+Gaussian charfun has purely real, negative exponent. -/
+private theorem integrable_charFun_gaussianVar (n : ℕ) :
+    Integrable (fun ξ => charFun (gaussianReal 0 (gaussianVar n)) ξ) volume := by
+  set v := gaussianVar n
+  have hv_pos : (0 : ℝ) < (v : ℝ≥0) := by
+    simp only [v, gaussianVar, NNReal.coe_mk]; positivity
+  -- The norm of charFun(N(0,v)) ξ equals exp(-(v/2)·ξ²)
+  have hnorm_eq : ∀ ξ : ℝ, ‖charFun (gaussianReal 0 v) ξ‖ = Real.exp (-((v : ℝ) / 2 * ξ ^ 2)) := by
+    intro ξ
+    rw [charFun_gaussianReal, Complex.norm_exp]
+    congr 1
+    simp only [ofReal_zero, mul_zero, zero_mul, zero_sub,
+      Complex.neg_re, Complex.div_ofNat_re, Complex.mul_re,
+      Complex.ofReal_re, Complex.ofReal_im, sub_zero]
+    rw [show ((↑ξ : ℂ) ^ 2).re = ξ ^ 2 by
+      simp only [sq, Complex.mul_re, Complex.ofReal_re, Complex.ofReal_im, mul_zero, sub_zero]]
+    ring
+  -- This function is integrable
+  refine (integrable_exp_neg_mul_sq (by linarith : 0 < (v : ℝ) / 2)).mono'
+    (continuous_charFun_gaussianVar n).aestronglyMeasurable ?_
+  exact ae_of_all _ fun ξ => by
+    rw [hnorm_eq]
+    exact Real.exp_le_exp_of_le (by nlinarith)
+
+/-- Integrability of the smoothed function (bounded by Gaussian). -/
+private theorem integrable_smoothed {φ : ℝ → ℂ} (hφc : Continuous φ)
+    (hpd : IsPositiveDefinite φ) (h0 : φ 0 = 1) (n : ℕ) :
+    Integrable (fun ξ => φ ξ * charFun (gaussianReal 0 (gaussianVar n)) ξ) volume := by
+  exact (integrable_charFun_gaussianVar n).mono
+    (continuous_smoothed hφc n).aestronglyMeasurable
+    (ae_of_all _ fun ξ => by
+      rw [norm_mul]
+      calc ‖φ ξ‖ * ‖charFun (gaussianReal 0 (gaussianVar n)) ξ‖
+          ≤ 1 * ‖charFun (gaussianReal 0 (gaussianVar n)) ξ‖ :=
+            mul_le_mul_of_nonneg_right (hpd.norm_le_one h0 ξ) (norm_nonneg _)
+        _ = ‖charFun (gaussianReal 0 (gaussianVar n)) ξ‖ := one_mul _)
+
+/-- The Gaussian charfun at a fixed `ξ` tends to 1 as the variance → 0. -/
+private theorem tendsto_gaussianCharFun_one (ξ : ℝ) :
+    Tendsto (fun n => charFun (gaussianReal 0 (gaussianVar n)) ξ) atTop (𝓝 1) := by
+  -- charFun(N(0,1/(n+1))) ξ → 1 as the Gaussian variance → 0
+  simp_rw [charFun_gaussianReal]
+  simp only [ofReal_zero, mul_zero, zero_mul, zero_sub]
+  -- Goal: cexp(-(↑↑(gaussianVar n) * ↑ξ^2/2)) → 1
+  rw [show (1 : ℂ) = cexp 0 from by simp]
+  apply Filter.Tendsto.cexp
+  -- The exponent -(v_n * ξ²/2) → 0 since v_n → 0
+  rw [show (0 : ℂ) = -(0 * (↑ξ : ℂ) ^ 2 / 2) from by simp]
+  -- Need: ↑↑(gaussianVar n) → 0 in ℂ
+  have hv_tendsto : Tendsto (fun n : ℕ => (↑↑(gaussianVar n) : ℂ)) atTop (𝓝 0) := by
+    rw [show (0 : ℂ) = ↑(0 : ℝ) from by simp]
+    exact (Complex.continuous_ofReal.tendsto 0).comp (by
+      show Tendsto (fun n : ℕ => (↑(gaussianVar n) : ℝ)) atTop (𝓝 0)
+      simp only [NNReal.coe_mk, gaussianVar]
+      exact tendsto_one_div_add_atTop_nhds_zero_nat)
+  exact (((hv_tendsto.mul tendsto_const_nhds).div_const _).neg)
+
+/-- The smoothed charfun converges pointwise to φ. -/
+private theorem tendsto_smoothed (φ : ℝ → ℂ) (ξ : ℝ) :
+    Tendsto (fun n => φ ξ * charFun (gaussianReal 0 (gaussianVar n)) ξ) atTop (𝓝 (φ ξ)) := by
+  conv_rhs => rw [show φ ξ = φ ξ * 1 from (mul_one _).symm]
+  exact tendsto_const_nhds.mul (tendsto_gaussianCharFun_one ξ)
+
+end GaussianSmoothing
+
+/-! ### Main theorem -/
+
 /-- **Bochner's theorem (ℝ only).** A continuous positive definite function `φ : ℝ → ℂ`
 with `φ(0) = 1` is the characteristic function of a unique probability measure on ℝ.
 
-Proof strategy (Gaussian smoothing + Lévy continuity):
-1. Define `φₙ(ξ) = φ(ξ) · exp(-ξ²/(2n))`. This is PD (Schur product with Gaussian charFun)
-   and L¹ (bounded × Gaussian decay).
-2. The Fourier transform `φ̂ₙ` is non-negative (from positive definiteness + Parseval).
-3. Since `φ̂ₙ ≥ 0` and `∫ φ̂ₙ = φₙ(0) = 1`, define `μₙ = φ̂ₙ · dx` as a probability measure.
-4. By construction, `charFun(μₙ) = φₙ → φ` pointwise as `n → ∞`.
-5. Apply Lévy's continuity theorem (`tendsto_of_charFunTendsto`) to extract the limit. -/
+Proof: Gaussian smoothing + Prokhorov + charfun identification. -/
 theorem bochner
     (φ : ℝ → ℂ) (hφc : Continuous φ) (hpd : IsPositiveDefinite φ)
     (h0 : φ 0 = 1) :
     ∃ μ : ProbabilityMeasure ℝ,
       ∀ ξ, MeasureTheory.ProbabilityMeasure.characteristicFun μ ξ = φ ξ := by
-  sorry
+  -- Step 1: For each n, the smoothed function φₙ(ξ) = φ(ξ) · charFun(N(0,1/(n+1)))(ξ)
+  -- is continuous, PD, L¹, and equals 1 at 0.
+  -- By exists_probMeasure_of_pd_integrable, get μₙ with charFun μₙ = φₙ.
+  have hsmoothed : ∀ n : ℕ, ∃ μn : ProbabilityMeasure ℝ,
+      ∀ ξ, ProbabilityMeasure.characteristicFun μn ξ =
+        φ ξ * charFun (gaussianReal 0 (gaussianVar n)) ξ :=
+    fun n => exists_probMeasure_of_pd_integrable _ (continuous_smoothed hφc n)
+      (pd_smoothed hpd n) (smoothed_at_zero h0 n) (integrable_smoothed hφc hpd h0 n)
+  -- Extract the sequence using choice.
+  choose μs hμs using hsmoothed
+  -- Step 2: The characteristic functions of μₙ converge pointwise to φ.
+  have hcharfun_conv :
+      ∀ ξ, Tendsto (fun n => charFun (μs n : Measure ℝ) ξ) atTop (𝓝 (φ ξ)) := by
+    intro ξ
+    have : ∀ n, charFun (μs n : Measure ℝ) ξ =
+        φ ξ * charFun (gaussianReal 0 (gaussianVar n)) ξ := by
+      intro n; exact (hμs n ξ).symm ▸ rfl
+    simp_rw [this]
+    exact tendsto_smoothed φ ξ
+  -- Step 3: The sequence {μₙ} is tight.
+  have htight := isTight_of_charFun_pointwise_tendsto hφc h0 hcharfun_conv
+  -- Step 4: Convert to the right form for Prokhorov.
+  have htight' : IsTightMeasureSet
+      {((ν : ProbabilityMeasure ℝ) : Measure ℝ) | ν ∈ range μs} := by
+    convert htight using 1
+    ext x; simp [mem_range]
+  -- Step 5: By Prokhorov, the closure of the range is compact.
+  have hcompact : IsCompact (closure (range μs)) :=
+    isCompact_closure_of_isTightMeasureSet htight'
+  -- Step 6: Extract a convergent subsequence.
+  have hin_closure : ∀ n, μs n ∈ closure (range μs) :=
+    fun n => subset_closure (mem_range_self n)
+  obtain ⟨ν, _, ns, hns_mono, hns_tendsto⟩ := hcompact.tendsto_subseq hin_closure
+  -- Step 7: By the easy direction of Lévy, charFun ν = lim charFun μₙₖ.
+  -- Also charFun μₙₖ → φ by our construction. By limit uniqueness, charFun ν = φ.
+  have hcharfun_ν : ∀ ξ, ProbabilityMeasure.characteristicFun ν ξ = φ ξ := by
+    intro ξ
+    -- charFun (μs (ns n)) ξ → charFun ν ξ (by weak convergence)
+    have h_weak := ProbabilityMeasure.charFunTendsto_of_tendsto hns_tendsto ξ
+    simp only [Function.comp_def, ProbabilityMeasure.characteristicFun_def] at h_weak
+    -- charFun (μs (ns n)) ξ → φ ξ (by our convergence + subsequence)
+    have h_phi := (hcharfun_conv ξ).comp hns_mono.tendsto_atTop
+    -- By uniqueness of limits
+    exact (ProbabilityMeasure.characteristicFun_def ν ξ).symm ▸
+      tendsto_nhds_unique h_weak h_phi
+  -- Step 8: ν is the desired probability measure.
+  exact ⟨ν, hcharfun_ν⟩
 
 end ProbabilityTheory

LeanLevy/Fourier/PositiveDefinite.lean

@@ -25,6 +25,10 @@ A function `φ : ℝ → ℂ` is **positive definite** if for every finite seque
 * `IsPositiveDefinite.mul` — Schur product: pointwise product of PD functions is PD.
 * `IsPositiveDefinite.of_charFun` — characteristic function of a probability measure is PD.
 * `IsPositiveDefinite.closure_pointwise` — pointwise limit of PD functions is PD.
+
+## Sorry audit
+
+* `norm_le_one` — PD bound `‖φ(ξ)‖ ≤ 1` when `φ(0) = 1` (2x2 PSD + Hermitian symmetry).
 -/
 
 open MeasureTheory Complex ComplexConjugate Finset Filter Topology Matrix
@@ -222,6 +226,15 @@ theorem closure_pointwise {φs : ℕ → ℝ → ℂ} (hφs : ∀ n, IsPositiveD
     exact (hlim (x i - x j)).const_mul _
   exact ge_of_tendsto htend (Eventually.of_forall fun k => hφs k n x c)
 
+/-- For a positive definite function with `φ(0) = 1`, `‖φ(ξ)‖ ≤ 1`.
+
+Proof sketch: Take `n = 2`, `x = (0, ξ)`, `c = (1, -φ(ξ)/‖φ(ξ)‖)`. The PD condition yields
+`0 ≤ 2 - 2‖φ(ξ)‖`, hence `‖φ(ξ)‖ ≤ 1`. Requires the Hermitian symmetry `φ(-ξ) = conj(φ(ξ))`
+which follows from the PD condition. -/
+theorem norm_le_one (hφ : IsPositiveDefinite φ) (h0 : φ 0 = 1) (ξ : ℝ) :
+    ‖φ ξ‖ ≤ 1 := by
+  sorry
+
 /-- The characteristic function of a probability measure is positive definite. -/
 theorem of_charFun (μ : ProbabilityMeasure ℝ) :
     IsPositiveDefinite (fun ξ => charFun (μ : Measure ℝ) ξ) := by