File size: 2,724 Bytes
c4cdbbd | 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 | import OAI.Probability.InvariantIsing.Spectral.PositiveSpectralLabels
import OAI.Probability.InvariantIsing.Spectral.FiniteSpectralMeasure
namespace OAI
/-! The positive part of a finite random label has the same distribution. -/
noncomputable section
open MeasureTheory Filter Set
open scoped BigOperators Classical
namespace InvariantIsing
variable {Ω A : Type*} [MeasurableSpace Ω] [Fintype A]
[MeasurableSpace A] [MeasurableSingletonClass A]
def finiteIndexWeight (P : Measure Ω) (g : Ω → A) (a : A) : ℝ :=
P.real (g ⁻¹' {a})
lemma finiteIndexWeight_sum (P : Measure Ω) [IsProbabilityMeasure P]
(g : Ω → A) (hg : Measurable g) : ∑ a, finiteIndexWeight P g a=1 := by
have hh := sum_measureReal_preimage_singleton (μ := P) Finset.univ
(f := g) (fun a _ => hg (measurableSet_singleton a))
simpa only [finiteIndexWeight,Finset.coe_univ,preimage_univ,probReal_univ] using hh
omit [MeasurableSpace A] [MeasurableSingletonClass A] in
lemma ae_finiteIndexWeight_pos (P : Measure Ω) [IsProbabilityMeasure P] (g : Ω → A) :
∀ᵐ x ∂P, 0 < finiteIndexWeight P g (g x) := by
have hi a : ∀ᵐ x ∂P, g x=a → 0 < finiteIndexWeight P g a := by
by_cases hp : 0 < finiteIndexWeight P g a
· exact Eventually.of_forall (fun _ _ => hp)
· have hz : P (g ⁻¹' {a})=0 := (measureReal_eq_zero_iff).mp
(le_antisymm (le_of_not_gt hp) measureReal_nonneg)
filter_upwards [compl_mem_ae_iff.mpr hz] with x hx
exact fun h => (hx h).elim
filter_upwards [ae_all_iff.mpr hi] with x hx
exact hx _ rfl
omit [MeasurableSpace A] [MeasurableSingletonClass A] in
lemma positiveIndexWeight (P : Measure Ω) [IsProbabilityMeasure P]
(g : Ω → A) (fallback : {a // 0 < finiteIndexWeight P g a})
(a : {a // 0 < finiteIndexWeight P g a}) :
finiteIndexWeight P (positiveSpectralLabel (finiteIndexWeight P g) fallback ∘ g) a=
finiteIndexWeight P g a := by
apply measureReal_congr
filter_upwards [ae_finiteIndexWeight_pos P g] with x hx
simp only [Function.comp_apply,positiveSpectralLabel,dite_eq_left hx,
mem_preimage,mem_singleton_iff]
exact propext Subtype.ext_iff
lemma finiteIndex_map (P : Measure Ω) [IsProbabilityMeasure P]
(g : Ω → A) (hg : Measurable g) (c : A → ℝ) :
P.map (c ∘ g)=finiteSpectralMeasure (finiteIndexWeight P g) c := by
have hc : Measurable c := measurable_of_finite c
rw [← Measure.map_map hc hg,Measure.map_eq_sum P g hg,Measure.map_sum hc.aemeasurable]
unfold finiteSpectralMeasure
congr 1
funext a
rw [Measure.map_smul _ hc.aemeasurable,Measure.map_dirac,finiteIndexWeight,
measureReal_def,ENNReal.ofReal_toReal (measure_ne_top P _)]
end InvariantIsing
end
end OAI
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