Pre-Brownian motion as a projective limit #
theorem
L2.posSemidef_interMatrix
{ι : Type u_1}
[Fintype ι]
{α : Type u_2}
{mα : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{v : ι → Set α}
(hv₁ : ∀ (j : ι), MeasurableSet (v j))
(hv₂ : ∀ (j : ι), μ (v j) ≠ ⊤ := by finiteness)
:
(Matrix.of fun (i j : ι) => μ.real (v i ∩ v j)).PosSemidef
theorem
ProbabilityTheory.integral_gaussianProjectiveFamily
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace ℝ E]
(I : Finset NNReal)
(f : ({ x : NNReal // x ∈ I } → ℝ) → E)
:
theorem
ProbabilityTheory.measurePreserving_gaussianProjectiveFamily
{I : Finset NNReal}
{s : { x : NNReal // x ∈ I }}
:
MeasureTheory.MeasurePreserving (fun (x : { x : NNReal // x ∈ I } → ℝ) => x s) (gaussianProjectiveFamily I)
(gaussianReal 0 ↑s)
theorem
MeasureTheory.IsProjectiveLimit.measurePreserving_restrict
{ι : Type u_2}
{X : ι → Type u_3}
{mX : (i : ι) → MeasurableSpace (X i)}
{μ : Measure ((i : ι) → X i)}
{P : (I : Finset ι) → Measure ((i : { x : ι // x ∈ I }) → X ↑i)}
(h : IsProjectiveLimit μ P)
{I : Finset ι}
:
MeasurePreserving I.restrict μ (P I)