File size: 3,438 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 69 70 71 72 73 74 75 | import OAI.Probability.InvariantIsing.Spectral.PositiveSpectralLabels
import OAI.Probability.InvariantIsing.Spectral.SpectralPartitionLaw
import Mathlib.MeasureTheory.Measure.Support
namespace OAI
/-! Positive-mass cells carry the same spectral law as the original partition. -/
noncomputable section
open MeasureTheory Set Filter
open scoped Topology Classical BigOperators Function
namespace InvariantIsing
lemma ae_spectralPartitionIndex_pos {ι : Type*} [Fintype ι]
(μ : Measure ℝ) [IsProbabilityMeasure μ] (S : ι → Set ℝ)
(hcover : ∀ x, ∃ i, x∈S i) :
∀ᵐ x ∂μ, 0 < μ.real (S (spectralPartitionIndex S hcover x)) := by
have hi (i : ι) : ∀ᵐ x ∂μ, x∈S i → 0 < μ.real (S i) := by
by_cases hp : 0 < μ.real (S i)
· exact Filter.Eventually.of_forall (fun _ _ => hp)
· have hz : μ (S i)=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 _ (spectralPartitionIndex_mem S hcover x)
lemma positive_spectral_partition_map {ι : Type*} [Fintype ι]
(μ : Measure ℝ) [IsProbabilityMeasure μ] (S : ι → Set ℝ)
(hcover : ∀ x, ∃ i, x∈S i) (hdis : Pairwise (Disjoint on S))
(hS : ∀ i, MeasurableSet (S i)) (lam : ι → ℝ)
(fallback : {i // 0 < μ.real (S i)}) :
μ.map (fun x => lam (positiveSpectralLabel (fun i => μ.real (S i)) fallback
(spectralPartitionIndex S hcover x)))=
finiteSpectralMeasure (fun i : {i // 0 < μ.real (S i)} => μ.real (S i))
(fun i => lam i) := by
have he : (fun x => lam (positiveSpectralLabel (fun i => μ.real (S i)) fallback
(spectralPartitionIndex S hcover x))) =ᵐ[μ]
(fun x => lam (spectralPartitionIndex S hcover x)) := by
filter_upwards [ae_spectralPartitionIndex_pos μ S hcover] with x hx
simp only [positiveSpectralLabel,dite_eq_left hx]
rw [Measure.map_congr he,spectralPartition_pushforward μ S hcover hdis hS lam,
finiteSpectralMeasure_positive _ _ (fun _ => measureReal_nonneg)]
lemma positive_spectral_edge_bounds {ι : Type*} [Fintype ι]
(μ : Measure ℝ) [IsProbabilityMeasure μ] (S : ι → Set ℝ)
(hcover : ∀ x, ∃ i, x∈S i) (lam : ι → ℝ) (δ a b : ℝ)
(ha : a∈μ.support) (hb : b∈μ.support)
(aMin aMax : {i // 0 < μ.real (S i)})
(hmin : ∀ i : {i // 0 < μ.real (S i)}, lam aMin ≤ lam i)
(hmax : ∀ i : {i // 0 < μ.real (S i)}, lam i ≤ lam aMax)
(hclose : ∀ᵐ x ∂μ, dist (lam (spectralPartitionIndex S hcover x)) x ≤ δ) :
lam aMin-δ ≤ a ∧ b ≤ lam aMax+δ := by
have hl : ∀ᵐ x ∂μ, x∈Ici (lam aMin-δ) := by
filter_upwards [ae_spectralPartitionIndex_pos μ S hcover,hclose] with x hx hc
have hh := hmin ⟨spectralPartitionIndex S hcover x,hx⟩
rw [Real.dist_eq,abs_le] at hc
change lam aMin-δ ≤ x
linarith
have hu : ∀ᵐ x ∂μ, x∈Iic (lam aMax+δ) := by
filter_upwards [ae_spectralPartitionIndex_pos μ S hcover,hclose] with x hx hc
have hh := hmax ⟨spectralPartitionIndex S hcover x,hx⟩
rw [Real.dist_eq,abs_le] at hc
change x ≤ lam aMax+δ
linarith
exact ⟨μ.support_subset_of_isClosed isClosed_Ici hl ha,
μ.support_subset_of_isClosed isClosed_Iic hu hb⟩
end InvariantIsing
end
end OAI
|