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f61e547 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 | import OAI.Probability.FreeIsing.Estimates.ProjectiveExtension
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367.BrownianExistence
theorem exists_preWienerMeasure :
∃ ν : Measure (ℝ≥0 → ℝ), IsProbabilityMeasure ν ∧
IsProjectiveLimit ν BrownianReal.projectiveFamily :=
CountableProjective.exists_projective_limit BrownianReal.projectiveFamily
BrownianReal.isProjectiveMeasureFamily_projectiveFamily
def preWienerMeasure : Measure (ℝ≥0 → ℝ) := Classical.choose exists_preWienerMeasure
instance : IsProbabilityMeasure preWienerMeasure :=
(Classical.choose_spec exists_preWienerMeasure).1
theorem preWienerMeasure_projectiveLimit :
IsProjectiveLimit preWienerMeasure BrownianReal.projectiveFamily :=
(Classical.choose_spec exists_preWienerMeasure).2
theorem coordinate_isPreBrownianReal :
IsPreBrownianReal (fun t (ω : ℝ≥0 → ℝ) => ω t) preWienerMeasure where
hasLaw I := {
aemeasurable := (Finset.measurable_restrict I).aemeasurable
map_eq := preWienerMeasure_projectiveLimit I }
end Problem367.BrownianExistence
end
end OAI
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