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a89d282 | 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 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 | import OAI.Probability.InvariantIsing.Spectral.CompactResolventTests
import OAI.Probability.InvariantIsing.Spectral.SpectralClipping
namespace OAI
/-! Clipped spectral probability laws on their compact interval. -/
noncomputable section
open MeasureTheory ProbabilityTheory Set
namespace InvariantIsing
def compactSpectralClip (R : ℝ) (hR : 0 ≤ R) (x : ℝ) : Icc (0 : ℝ) R :=
⟨spectralClip 0 R x,spectralClip_mem hR x⟩
lemma continuous_compactSpectralClip (R : ℝ) (hR : 0 ≤ R) : Continuous (compactSpectralClip R hR) :=
(continuous_spectralClip 0 R).subtype_mk _
def compactClippedLaw (R : ℝ) (hR : 0 ≤ R) (μ : ProbabilityMeasure ℝ) :
ProbabilityMeasure (Icc (0 : ℝ) R) := μ.map (compactSpectralClip R hR)
lemma compactClippedLaw_real (R : ℝ) (hR : 0 ≤ R) (μ : ProbabilityMeasure ℝ) :
(compactClippedLaw R hR μ).map (fun x => (x : ℝ)) = μ.map (spectralClip 0 R) := by
apply ProbabilityMeasure.toMeasure_injective
simp only [compactClippedLaw,ProbabilityMeasure.toMeasure_map]
rw [Measure.map_map measurable_subtype_coe (continuous_compactSpectralClip R hR).measurable]
rfl
lemma compactClippedLaw_integral (R : ℝ) (hR : 0 ≤ R) (μ : ProbabilityMeasure ℝ)
(t : {t : ℝ // 0 < t}) :
(∫ x, compactResolventTest R t x ∂(compactClippedLaw R hR μ : Measure _)) =
∫ x, positiveResolventTest t (spectralClip 0 R x) ∂(μ : Measure ℝ) := by
rw [compactClippedLaw,ProbabilityMeasure.toMeasure_map,
integral_map (continuous_compactSpectralClip R hR).measurable.aemeasurable
(compactResolventTest R t).continuous.aestronglyMeasurable]
rfl
lemma compactClippedLaw_real_eq (R : ℝ) (hR : 0 ≤ R) (μ : ProbabilityMeasure ℝ)
(hμ : ∀ᵐ x ∂(μ : Measure ℝ), x ∈ Icc (0 : ℝ) R) :
(compactClippedLaw R hR μ).map (fun x => (x : ℝ)) = μ := by
rw [compactClippedLaw_real]
apply ProbabilityMeasure.toMeasure_injective
rw [ProbabilityMeasure.toMeasure_map]
calc
(μ : Measure ℝ).map (spectralClip 0 R) = (μ : Measure ℝ).map id :=
Measure.map_congr (hμ.mono fun _ hx => spectralClip_eq hx)
_ = μ := Measure.map_id
end InvariantIsing
end
end OAI
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