[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"qJqleH8jwM":3},"# Bochner's and Minlos' Theorems in Lean 4\n\nFormal proofs of **Bochner's theorem** and **Minlos' theorem** in Lean 4 — the two foundational results characterizing when a function(al) is the Fourier transform of a probability measure — using [Mathlib](https://github.com/leanprover-community/mathlib4) and [KolmogorovExtension4](https://github.com/remydegenne/kolmogorov_extension4).\n\n## Bochner's theorem\n\n**Statement**: A continuous function φ : V → ℂ on a finite-dimensional real inner product space is positive definite with φ(0) = 1 if and only if it is the characteristic function of a unique probability measure on V.\n\n**Status**: 0 sorries, 0 axioms — fully proved! `#print axioms bochner_theorem` shows only standard Lean axioms.\n\nThe hard direction (PD implies measure) uses **Gaussian regularization**, avoiding Riesz-Markov-Kakutani:\n\n1. **Fourier positivity**: For φ continuous, integrable, and positive-definite, Re(𝓕φ(ξ)) ≥ 0 for all ξ. Proved via Fejér-averaged double integrals.\n2. **Gaussian regularization**: φ_ε(x) = φ(x) · exp(-ε‖x‖²) is PD (Schur product theorem) with Gaussian decay, hence L¹.\n3. **Measure construction**: 𝓕φ_ε ≥ 0 gives dμ_ε = 𝓕φ_ε dλ, a probability measure with charFun(μ_ε) = φ_ε.\n4. **Weak limit**: Prokhorov's theorem extracts a weakly convergent subsequence as ε → 0; the limit measure has charFun = φ.\n5. **Uniqueness**: From Mathlib's `Measure.ext_of_charFun`.\n\n## Minlos' theorem\n\n**Statement** (Lean):\n```lean\ntheorem minlos_theorem [IsHilbertNuclear E] [SeparableSpace E] [Nonempty E]\n    (Φ : E → ℂ) (h_continuous : Continuous Φ)\n    (h_positive_definite : IsPositiveDefinite Φ) (h_normalized : Φ 0 = 1) :\n    ∃! μ : ProbabilityMeasure (WeakDual ℝ E),\n      ∀ f : E, Φ f = ∫ ω, exp (I * (ω f)) ∂μ.toMeasure\n```\n\nOn a nuclear, separable, locally convex space E, a continuous positive-definite normalized functional Φ : E → ℂ is the characteristic functional of a unique probability measure on the topological dual E' = WeakDual ℝ E.\n\n**Status**: 0 sorries, 0 axioms — fully proved! `#print axioms minlos_theorem` shows only standard Lean axioms.\n\n**Proof strategy**:\n1. **Finite-dimensional marginals**: For each finite set {f₁,...,fₙ} ⊂ E, apply Bochner's theorem to get a measure μ_F on ℝⁿ.\n2. **Projective family**: Transport marginals to `∀ j : J, ℝ` forming an `IsProjectiveMeasureFamily`.\n3. **Kolmogorov extension**: The `projectiveLimit` (imported from KolmogorovExtension4) gives a measure ν on E → ℝ.\n4. **Measurable modification**: Push forward ν through a measurable map P : (E → ℝ) → WeakDual ℝ E.\n5. **CF verification**: P(ω)(f) = ω(f) ν-a.e., so charFun(μ) = Φ.\n6. **Uniqueness**: P ∘ embed = id implies the representing measure is unique.\n\n## Nuclear spaces\n\nThe project formalizes two characterizations of nuclear spaces and proves the Pietsch characterization implies the Hilbertian one. The [gaussian-field](https://github.com/YulinGu-Fly/OSforGFF) project independently defines `NuclearSpace` using the same Pietsch characterization; our `IsNuclear` matches it, and the bridge `isHilbertNuclear_of_nuclear` completes the chain to `IsHilbertNuclear` used by Minlos' theorem.\n\n### Hilbertian characterization (used by Minlos)\n\nA locally convex space is **nuclear** if its topology is generated by a countable family of Hilbertian seminorms (satisfying the parallelogram law) with Hilbert-Schmidt inclusions between consecutive levels:\n\n```lean\nclass IsHilbertNuclear (E : Type*) [AddCommGroup E] [Module ℝ E]\n    [TopologicalSpace E] : Prop where\n  nuclear_hilbert_embeddings :\n    ∃ (p : ℕ → Seminorm ℝ E),\n      (∀ n, (p n).IsHilbertian) ∧\n      (WithSeminorms (fun n => p n)) ∧\n      (∀ n, (p (n + 1)).IsHilbertSchmidtEmbedding (p n))\n```\n\nThis is the characterization from Gel'fand-Vilenkin and Trèves (Ch. 50). The Hilbert-Schmidt condition means the sum ∑ q(eₖ)² is uniformly bounded over all finite p-orthonormal sequences {eₖ}.\n\n### Pietsch characterization\n\nEvery continuous seminorm p is dominated by a **nuclear expansion**: p(x) ≤ Σₙ |fₙ(x)| · cₙ where the fₙ are continuous linear functionals bounded by a larger seminorm q, and Σ cₙ \u003C ∞:\n\n```lean\ndef IsNuclear (E : Type*) [AddCommGroup E] [Module ℝ E]\n    [TopologicalSpace E] : Prop :=\n  ∀ (p : Seminorm ℝ E), Continuous p →\n    ∃ (q : Seminorm ℝ E), Continuous q ∧ (∀ x, p x ≤ q x) ∧\n    ∃ (f : ℕ → (E →L[ℝ] ℝ)) (c : ℕ → ℝ),\n      (∀ n, 0 ≤ c n) ∧ Summable c ∧\n      (∀ n x, |f n x| ≤ q x) ∧\n      (∀ x, p x ≤ ∑' n, |f n x| * c n)\n```\n\nThis is Pietsch's original definition, equivalent to requiring that the canonical maps between seminorm completions are nuclear operators.\n\n### Bridge: Pietsch implies Hilbertian\n\n```lean\ntheorem isHilbertNuclear_of_nuclear [IsTopologicalAddGroup E] [ContinuousSMul ℝ E]\n    (hPN : IsNuclear E)\n    (q₀ : ℕ → Seminorm ℝ E) (hq₀ : WithSeminorms q₀)\n    (hq₀_cont : ∀ n, Continuous (q₀ n)) :\n    IsHilbertNuclear E\n```\n\nThe bridge constructs Hilbertian seminorms via a **Hilbertian lift** r(x) = √(Σₖ fₖ(x)² · cₖ) from each nuclear expansion, then shows the inclusions are Hilbert-Schmidt using a Bessel inequality argument. **Fully proved** (0 sorries, 0 axioms).\n\n## Project structure\n\nThree Lean libraries (~8300 lines, 0 sorries, 0 custom axioms):\n\n### Bochner — `Bochner/`\n\nBochner's theorem and supporting infrastructure.\n\n| File | Contents |\n|------|----------|\n| [PositiveDefinite](Bochner/PositiveDefinite.lean) | [IsPositiveDefinite, Schur product theorem, precomp with linear maps](summary/Bochner/PositiveDefinite.md) |\n| [FejerPD](Bochner/FejerPD.lean) | [Fourier positivity: Re(𝓕φ) ≥ 0 via Fejér averaging](summary/Bochner/FejerPD.md) |\n| [Main](Bochner/Main.lean) | [Bochner's theorem: Gaussian regularization, Prokhorov, uniqueness](summary/Bochner/Main.md) |\n| [Sazonov](Bochner/Sazonov.lean) | [Sazonov topology via trace-class seminorms](summary/Bochner/Sazonov.md) |\n| [TestFubini](Bochner/TestFubini.lean) | [Fubini lemmas for Gaussian regularization integrals](summary/Bochner/TestFubini.md) |\n\n---\n\n### Minlos — `Minlos/`\n\nMinlos' theorem, nuclear spaces, and concentration bounds.\n\n| File | Contents |\n|------|----------|\n| [NuclearSpace](Minlos/NuclearSpace.lean) | [IsHilbertNuclear class, Hilbertian seminorms, HS embeddings](summary/Minlos/NuclearSpace.md) |\n| [PietschBridge](Minlos/PietschBridge.lean) | [IsNuclear → IsHilbertNuclear via hilbertianLift + Bessel inequality](summary/Minlos/PietschBridge.md) |\n| [SazonovTightness](Minlos/SazonovTightness.lean) | [Sazonov CF continuity → tightness of finite-dim marginals](summary/Minlos/SazonovTightness.md) |\n| [MinlosConcentration](Minlos/MinlosConcentration.lean) | [Concentration theorem: Gaussian averaging, Gram matrix, Chebyshev](summary/Minlos/MinlosConcentration.md) |\n| [FinDimMarginals](Minlos/FinDimMarginals.lean) | [Finite-dimensional marginal measures via Bochner's theorem](summary/Minlos/FinDimMarginals.md) |\n| [ProjectiveFamily](Minlos/ProjectiveFamily.lean) | [Kolmogorov projective family + extension to E → ℝ](summary/Minlos/ProjectiveFamily.md) |\n| [MeasurableModification](Minlos/MeasurableModification.lean) | [Measurable projection P : (E → ℝ) → WeakDual ℝ E](summary/Minlos/MeasurableModification.md) |\n| [Main](Minlos/Main.lean) | [Minlos' theorem: existence + uniqueness on nuclear spaces](summary/Minlos/Main.md) |\n\n---\n\n### Test — `Test/`\n\nIntegration tests using axioms for Schwartz space properties. The [gaussian-field](https://github.com/YulinGu-Fly/OSforGFF) project proves that Schwartz space is nuclear (via Hermite–Sobolev norms and the Dynin-Mityagin characterization); here we axiomatize this and other Schwartz properties to demonstrate Minlos' theorem constructing white noise on S'(ℝ).\n\n| File | Contents |\n|------|----------|\n| [WhiteNoise](Test/WhiteNoise.lean) | [White noise measure on S'(ℝ) via Minlos, bridge theorem with gaussian-field](summary/Test/WhiteNoise.md) |\n\n## Building\n\n```bash\nlake update\nlake build\n```\n\nRequires Lean v4.28.0-rc1 and Mathlib.\n\n## Key Mathlib dependencies\n\n- `Mathlib.Analysis.Fourier.FourierTransform` — Fourier transform on inner product spaces\n- `Mathlib.Analysis.Fourier.Inversion` — Fourier inversion theorem\n- `Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform` — Gaussian Fourier transform\n- `Mathlib.MeasureTheory.Measure.Prokhorov` — Prokhorov's theorem\n- `Mathlib.MeasureTheory.Measure.CharacteristicFunction` — Characteristic functions of measures\n- `Mathlib.LinearAlgebra.Matrix.PosDef` — Positive semidefinite matrices\n- `Mathlib.Analysis.LocallyConvex.WithSeminorms` — Seminorm-generated topologies\n- `Mathlib.Topology.Algebra.Module.WeakDual` — Weak-* dual spaces\n- [KolmogorovExtension4](https://github.com/remydegenne/kolmogorov_extension4) — Kolmogorov extension theorem (projective limits of measures)\n\n## References\n\n- Rudin, *Fourier Analysis on Groups*, Theorem 1.4.3\n- Folland, *A Course in Abstract Harmonic Analysis*, Chapter 4\n- Reed and Simon, *Methods of Modern Mathematical Physics I*, Section IX.9\n- Minlos, *Generalized random processes and their extension to a measure* (1959)\n- Gel'fand and Vilenkin, *Generalized Functions* Vol. 4, Chapters 3-4\n- Trèves, *Topological Vector Spaces*, Chapters 50-51\n- Pietsch, *Nuclear Locally Convex Spaces* (1972)\n- Degenne, Ledvinka, Marion, Pfaffelhuber, *Formalization of Brownian motion in Lean* (2025), [arXiv:2511.20118](https://arxiv.org/abs/2511.20118)\n\n## Author\n\nMichael R. Douglas\n\n## License\n\nCopyright (c) 2026 Michael R. Douglas. Released under the Apache 2.0 license.\n",1780846776851]