File size: 3,033 Bytes
135052a | 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 | import Mathlib
import OAI.Probability.SKBarriers.Parisi.CDFSupportContact
import OAI.Probability.SKBarriers.Parisi.CDFHeatContact
namespace OAI
section
noncomputable section
open scoped NNReal Topology BigOperators
open MeasureTheory ProbabilityTheory Filter Set
namespace SK.Analytic
theorem cdf_zero_below_least_support (μ : ProbabilityMeasure ℝ) {q z : ℝ}
(hq : IsLeast (μ : Measure ℝ).support q) (hz : z<q) :
cdf (μ : Measure ℝ) z=0 := by
have H : (μ : Measure ℝ) (Iic z)=0 := by
apply measure_mono_null (t:=(μ : Measure ℝ).supportᶜ) _ (μ : Measure ℝ).measure_compl_support
intro x hx hs
exact (not_le_of_gt hz) (le_trans (hq.2 hs) hx)
rw [cdf_eq_real,measureReal_def,H,ENNReal.toReal_zero]
theorem scalarCDFParisi_zero_mem_support {β : ℝ} (hβ : β≠0)
(μ : ProbabilityMeasure ℝ) (hμ : (μ : Measure ℝ) (Icc (0:ℝ) 1)=1)
(hmin : scalarCDFParisi β (cdf (μ : Measure ℝ))=finiteParisiInf β) :
(0:ℝ)∈(μ : Measure ℝ).support := by
have HS := Measure.support_subset_of_isClosed (μ:=(μ : Measure ℝ))
isClosed_Icc (supported_probability_ae μ hμ)
have HC : IsCompact (μ : Measure ℝ).support :=
isCompact_Icc.of_isClosed_subset (μ : Measure ℝ).isClosed_support HS
obtain ⟨q,hq⟩ := HC.exists_isLeast ((μ : Measure ℝ).nonempty_support (by
intro H
have H1 := measure_univ (μ:=(μ : Measure ℝ))
simp only [H,Measure.coe_zero,Pi.zero_apply,zero_ne_one] at H1))
have hbound := HS hq.1
have he : q=0 := by
by_contra H
have hq0 : 0<q := lt_of_le_of_ne hbound.1 (Ne.symm H)
have hm := (scalarCDFParisi_support_minima hβ μ hμ hmin q hq.1).2.2
exact scalarCDFPotential_not_min_heat_prefix β (cdf (μ : Measure ℝ))
(fun z => ⟨cdf_nonneg _ z,cdf_le_one _ z⟩) (supported_probability_cdf_one μ hμ)
hq0 hbound.2 (fun z hz => cdf_zero_below_least_support μ hq hz.2)
(scalarCDFParisi_overlap_on_support hβ μ hμ hmin hq.1) hm
simpa only [he] using hq.1
theorem cdf_pos_of_zero_mem_support (μ : ProbabilityMeasure ℝ)
(h0 : (0:ℝ)∈(μ : Measure ℝ).support) {s : ℝ} (hs : 0<s) :
0<cdf (μ : Measure ℝ) s := by
have H : 0<(μ : Measure ℝ) (Iic s) :=
(Measure.mem_support_iff_forall (μ:=(μ : Measure ℝ)) 0).mp h0 _ (Iic_mem_nhds hs)
rw [cdf_eq_real,measureReal_def]
exact ENNReal.toReal_pos H.ne' (measure_ne_top _ _)
theorem cdf_area_strict (μ : ProbabilityMeasure ℝ)
(h0 : (0:ℝ)∈(μ : Measure ℝ).support) {h : ℝ} (hh : 0<h) :
(∫ s in h..1, cdf (μ : Measure ℝ) s)<(∫ s in (0:ℝ)..1, cdf (μ : Measure ℝ) s) := by
have HI (a b : ℝ) : IntervalIntegrable (cdf (μ : Measure ℝ)) volume a b :=
(cdf (μ : Measure ℝ)).mono.intervalIntegrable
have HP := intervalIntegral.intervalIntegral_pos_of_pos_on (HI 0 h)
(fun s hs => cdf_pos_of_zero_mem_support μ h0 hs.1) hh
have HE := intervalIntegral.integral_add_adjacent_intervals (HI 0 h) (HI h 1)
linarith
end SK.Analytic
end
end
end OAI
|