File size: 3,396 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 | import OAI.Probability.InvariantIsing.Pressure.RandomOrbitIntegration
import OAI.Probability.InvariantIsing.Pressure.UniformPhysicalTail
namespace OAI
/-! A localized tail bound under the exact joint conditional orbit hypothesis. -/
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal
namespace InvariantIsing
lemma conditional_orbit_tail_bound {Ω : Type*} [MeasurableSpace Ω] {N : ℕ}
(hN : 0 < N) (P : Measure Ω) [IsProbabilityMeasure P]
(d : Ω → FieldSpectralData N) (Y : Ω → ℝ) (hd : Measurable d) (hY : Measurable Y)
(H : Measure (Orthogonal N)) [IsProbabilityMeasure H]
(hlaw : ConditionalFieldOrbitLaw P d Y H) (K ε B : ℝ)
(hbound : ∀ data : FieldSpectralData N, (∀ i, |data.1 i| ≤ K) →
H.real {U | ε ≤ |dataPhysicalPressure data U-(∫ V, dataPhysicalPressure data V ∂H)|} ≤ B) :
P {ω | (∀ i, |(d ω).1 i| ≤ K) ∧
ε ≤ |Y ω-(∫ V, dataPhysicalPressure (d ω) V ∂H)|} ≤ ENNReal.ofReal B := by
let mean := fun data : FieldSpectralData N => ∫ V, dataPhysicalPressure data V ∂H
have hm : Measurable mean := measurable_dataPressureMean hN H
let E : Set (FieldSpectralData N × ℝ) := {z | (∀ i, |z.1.1 i| ≤ K) ∧ ε ≤ |z.2-mean z.1|}
have hK : MeasurableSet {z : FieldSpectralData N × ℝ | ∀ i, |z.1.1 i| ≤ K} := by
simp only [ofPred_forall]
apply MeasurableSet.iInter
intro i
exact measurableSet_le (((measurable_pi_apply i).comp
(measurable_fst.comp measurable_fst)).abs) measurable_const
have hE : MeasurableSet E := hK.inter
(measurableSet_le measurable_const (measurable_snd.sub (hm.comp measurable_fst)).abs)
have hp : Measurable (fun z : FieldSpectralData N × Orthogonal N =>
(z.1,dataPhysicalPressure z.1 z.2)) :=
measurable_fst.prodMk (measurable_dataPhysicalPressure hN)
have he := congrArg (fun Q : Measure (FieldSpectralData N × ℝ) => Q E) hlaw
rw [Measure.map_apply (hd.prodMk hY) hE,Measure.map_apply hp hE,
Measure.prod_apply (hp hE)] at he
let : IsProbabilityMeasure (P.map d) :=
(Measure.isProbabilityMeasure_map_iff hd.aemeasurable).mpr inferInstance
change P ((fun ω => (d ω,Y ω)) ⁻¹' E) ≤ _
rw [he]
calc
_ ≤ ∫⁻ _data : FieldSpectralData N, ENNReal.ofReal B ∂P.map d := by
apply lintegral_mono
intro data
dsimp only
by_cases hdata : ∀ i, |data.1 i| ≤ K
· have hevent : Prod.mk data ⁻¹' ((fun z : FieldSpectralData N × Orthogonal N =>
(z.1,dataPhysicalPressure z.1 z.2)) ⁻¹' E)=
{U | ε ≤ |dataPhysicalPressure data U-mean data|} := by
ext U
change ((∀ i, |data.1 i| ≤ K) ∧ ε ≤ |dataPhysicalPressure data U-mean data|) ↔ _
exact and_iff_right hdata
rw [hevent]
have hb := ENNReal.ofReal_le_ofReal (hbound data hdata)
rwa [measureReal_def,ENNReal.ofReal_toReal (measure_ne_top H _)] at hb
· have hevent : Prod.mk data ⁻¹' ((fun z : FieldSpectralData N × Orthogonal N =>
(z.1,dataPhysicalPressure z.1 z.2)) ⁻¹' E)=∅ := by
ext U
change ((∀ i, |data.1 i| ≤ K) ∧ ε ≤ |dataPhysicalPressure data U-mean data|) ↔ False
simp only [hdata,false_and]
rw [hevent,measure_empty]
exact bot_le
_ = _ := by simp
end InvariantIsing
end
end OAI
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