File size: 2,979 Bytes
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 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 | import OAI.Probability.InvariantIsing.Spectral.MeasureResolvent
namespace OAI
/-! The general measure transform agrees with the finite-alphabet transform.
Positive masses ensure that the upper edge is an atom and the inverse is uncapped. -/
noncomputable section
open MeasureTheory Filter Set
open scoped BigOperators
namespace InvariantIsing
variable {ι : Type*} [Fintype ι]
def finiteSpectralMeasure (ρ eig : ι → ℝ) : Measure ℝ :=
Measure.sum fun a => ENNReal.ofReal (ρ a) • Measure.dirac (eig a)
theorem finiteSpectralMeasure_probability (ρ eig : ι → ℝ)
(hρ : ∀ a, 0 ≤ ρ a) (hsum : ∑ a, ρ a = 1) :
IsProbabilityMeasure (finiteSpectralMeasure ρ eig) := by
apply HasSum.isProbabilityMeasure_sum_dirac hρ
simpa only [hsum] using hasSum_fintype ρ
theorem integral_finiteSpectralMeasure (ρ eig : ι → ℝ) (hρ : ∀ a, 0 ≤ ρ a)
(f : ℝ → ℝ) :
(∫ y, f y ∂finiteSpectralMeasure ρ eig) = ∑ a, ρ a * f (eig a) := by
have hi a : Integrable f (ENNReal.ofReal (ρ a) • Measure.dirac (eig a)) :=
(integrable_dirac (by simp)).smul_measure (by finiteness)
rw [finiteSpectralMeasure, Measure.sum_fintype,
integral_finsetSum_measure (fun a _ => hi a)]
simp only [integral_smul_measure, integral_dirac, ENNReal.toReal_ofReal (hρ _), smul_eq_mul]
theorem ae_finiteSpectralMeasure (ρ eig : ι → ℝ) {P : ℝ → Prop} (hP : ∀ a, P (eig a)) :
∀ᵐ y ∂finiteSpectralMeasure ρ eig, P y := by
rw [finiteSpectralMeasure, Measure.ae_sum_iff]
intro a
have hd : ∀ᵐ y ∂Measure.dirac (eig a), P y := by
simpa only [ae_dirac_eq, Filter.eventually_pure] using hP a
exact Measure.ae_smul_measure hd _
theorem measureResolvent_finiteSpectralMeasure (ρ eig : ι → ℝ) (hρ : ∀ a, 0 ≤ ρ a)
(b : ℝ) : measureResolvent (finiteSpectralMeasure ρ eig) b = finiteResolvent ρ eig b := by
rw [measureResolvent, integral_finiteSpectralMeasure ρ eig hρ]
simp only [finiteResolvent, mul_one_div]
theorem measureR_finiteSpectralMeasure (ρ eig : ι → ℝ)
(hρ : ∀ a, 0 < ρ a) (hsum : ∑ a, ρ a = 1)
(aMax : ι) (hMax : ∀ a, eig a ≤ eig aMax) (x : ℝ) :
measureR (finiteSpectralMeasure ρ eig) (eig aMax) x = finiteR ρ eig hρ hsum x := by
let := finiteSpectralMeasure_probability ρ eig (fun a => (hρ a).le) hsum
by_cases hx : 0 < x
· have hs := finiteInverse_spec ρ eig hρ hsum hx
have hsol : measureResolvent (finiteSpectralMeasure ρ eig) (finiteInverse ρ eig hρ hsum x) = x := by
rw [measureResolvent_finiteSpectralMeasure ρ eig (fun a => (hρ a).le)]
exact hs.2
have hi := measureInverse_eq_of_solution (finiteSpectralMeasure ρ eig)
(ae_finiteSpectralMeasure ρ eig hMax) (hs.1 aMax) hsol
simp only [measureR, finiteR, ite_eq_left hx, hi]
· simp only [measureR, finiteR, ite_eq_right hx,
integral_finiteSpectralMeasure ρ eig (fun a => (hρ a).le)]
end InvariantIsing
end
end OAI
|