Samplinglib mathematical lesson

The full orthogonal group has a unique Haar probability

AutoSamplingTheory.TechnicalLemmas.Probability.OrthogonalHaar.existsUnique_orthogonalHaar · theorem · Teaching coverage

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Mathematical statement

For every natural finite dimension n, the full real orthogonal group O(n), with its inherited matrix topology and Borel sigma-algebra, has exactly one measure which is both a probability and a Haar measure. This explicates the standard Haar background invoked in the source.

\[\exists!\,\mu\in\mathcal M(O(n)):\quad \mu(O(n))=1\ \land\ \mu\text{ is Haar}\]

Lean statement · existsUnique_orthogonalHaar

The measure lives on the actual full orthogonal subtype with its canonical Borel space. No abstract Haar law or compactness premise is supplied by the caller.

Braces mark parameters Lean can infer; square brackets request structures such as a measurable space or probability measure. Named hypotheses are mathematical premises, not facts established by this declaration. Section parameters are described in the mathematical hypotheses above; the module link retains their exact source context.

Download module (.lean)

theorem existsUnique_orthogonalHaar (n : ℕ) :
    ∃! μ : Measure (Matrix.orthogonalGroup (Fin n) ℝ),
      IsProbabilityMeasure μ ∧ μ.IsHaarMeasure

Exact module and namespace context

Proof roadmap

1The group is compact in its actual matrix topology
Orthogonality bounds every real matrix entry in absolute value by one. The unitary set is closed, so O(n) is a closed subset of the finite product cube [-1,1]^(n times n). The cube is compact. Identity supplies nonemptiness even at n=0; the subspace is second countable and has its canonical Borel structure.
2Normalize and identify the unique law
Apply the actual Mathlib Haar constructor with positive compact normalization set K0 equal to the whole compact group. The constructor assigns K0 mass one. Haar uniqueness on the second-countable group shows that any other Haar probability is equal to this one.

Full assumptions and notation

All objects and hypotheses

n is an arbitrary natural finite dimension. The paper applications use n≥3 or d≥2; the shared background also covers n=0.
O(n) is the full real orthogonal group O^T O=I, equipped with the subspace topology of finite real matrices and its Borel sigma-algebra. Both determinant signs are allowed whenever they exist.
Haar means left invariant, finite on compact sets and positive on nonempty open sets. Probability means total mass one. Compactness and second countability are proved internally, not assumed as public premises.

Mathematical proof

1. The group is compact in its actual matrix topology

Orthogonality bounds every real matrix entry in absolute value by one. The unitary set is closed, so O(n) is a closed subset of the finite product cube [-1,1]^(n times n). The cube is compact. Identity supplies nonemptiness even at n=0; the subspace is second countable and has its canonical Borel structure.

\[O(n)\subseteq[-1,1]^{n\times n},\qquad O(n)\text{ closed}\ \Longrightarrow\ O(n)\text{ compact}\]

Corresponding Lean step

Private instances construct the canonical Borel space, induced second countability and compactness from entry_norm_bound_of_unitary, isClosed_unitary and isCompact_Icc. They do not add source assumptions.

2. Normalize and identify the unique law

Apply the actual Mathlib Haar constructor with positive compact normalization set K0 equal to the whole compact group. The constructor assigns K0 mass one. Haar uniqueness on the second-countable group shows that any other Haar probability is equal to this one.

\[\mu(K_0)=1,\qquad \nu,\mu\text{ Haar probabilities}\ \Longrightarrow\nu=\mu\]

Corresponding Lean step

haarMeasure_self and PositiveCompacts.coe_top give probability; the constructor supplies IsHaarMeasure; isHaarMeasure_eq_of_isProbabilityMeasure gives equality for every competing measure.

Lean verification

The exact proof begins folded immediately after the mathematical argument; the statement is folded beside the claim above.

Lean proof · existsUnique_orthogonalHaar

The adjacent exact Lean constructs the law and discharges its topology, normalization and invariance dependencies internally.

Download module (.lean)

theorem existsUnique_orthogonalHaar (n : ℕ) :
    ∃! μ : Measure (Matrix.orthogonalGroup (Fin n) ℝ),
      IsProbabilityMeasure μ ∧ μ.IsHaarMeasure := by
  let μ : Measure (Matrix.orthogonalGroup (Fin n) ℝ) :=
    Measure.haarMeasure (⊤ : PositiveCompacts (Matrix.orthogonalGroup (Fin n) ℝ))
  have hprob : IsProbabilityMeasure μ := ⟨by
    simpa only [μ, PositiveCompacts.coe_top] using
      (Measure.haarMeasure_self (K₀ :=
        (⊤ : PositiveCompacts (Matrix.orthogonalGroup (Fin n) ℝ))))⟩
  have hhaar : μ.IsHaarMeasure := by dsimp only [μ]; infer_instance
  refine ⟨μ, ⟨hprob, hhaar⟩, ?_⟩
  intro ν hν
  let := hprob
  let := hhaar
  let := hν.1
  let := hν.2
  exact Measure.isHaarMeasure_eq_of_isProbabilityMeasure ν μ

Exact module and namespace context

Scope and omitted-condition boundaries

Only the common full orthogonal Haar probability and fixed left multiplication invariance. Determinant-conditioned laws, component half-mass, principal-minor sign/zero expectation, Gaussian densities, independence from an algorithm seed, adaptive posterior invariance, Krylov reduction and both main lower bounds remain open. Dimension zero is included as a singleton group; no two-components claim is made there.

Sources, dependencies, and reuse

Source and reuse

Samplinglib parents called

Mathlib API called (external library)

entry_norm_bound_of_unitary
isClosed_unitary
isCompact_Icc
MeasureTheory.Measure.haarMeasure_self
MeasureTheory.Measure.isHaarMeasure_eq_of_isProbabilityMeasure

Mathematical sources

Shared full Haar background — Background common to both experiments; only the fixed left-invariance part of the row-reflection argument.

Samplinglib prose is not a quotation or a source-equivalence certificate. Definitions and aliases are explained as constructions, not counted as new mathematical proofs.