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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 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 | import OAI.Probability.InvariantIsing.Spectral.MeasureResolvent
namespace OAI
/-! Order tests and continuity for the inverse across the capped branch. -/
noncomputable section
open MeasureTheory Filter Set
open scoped Topology Classical
namespace InvariantIsing
theorem measureInverse_le_iff (μ : Measure ℝ) [IsProbabilityMeasure μ]
{e x b : ℝ} (hμ : ∀ᵐ y ∂μ, y ≤ e) (hx : 0 < x) (hb : e < b) :
measureInverse μ e x ≤ b ↔ measureResolvent μ b ≤ x := by
by_cases hex : ∃ r, e < r ∧ measureResolvent μ r = x
· rw [measureInverse, dite_eq_left hex]
have hs := hex.choose_spec
constructor
· intro hrb
rcases hrb.eq_or_lt with h | h
· rw [← h, hs.2]
· have ht := measureResolvent_strictAnti μ hμ hs.1 h
rw [hs.2] at ht
exact ht.le
· intro hR
by_contra hnot
have ht := measureResolvent_strictAnti μ hμ hb (lt_of_not_ge hnot)
rw [hs.2] at ht
exact (not_lt_of_ge hR) ht
· rw [measureInverse, dite_eq_right hex]
constructor
· intro _
by_contra hnot
exact hex (exists_measureResolvent_solution μ hμ hb hx (lt_of_not_ge hnot).le)
· intro _
exact hb.le
theorem measureInverse_lt_iff (μ : Measure ℝ) [IsProbabilityMeasure μ]
{e x b : ℝ} (hμ : ∀ᵐ y ∂μ, y ≤ e) (hx : 0 < x) (hb : e < b) :
measureInverse μ e x < b ↔ measureResolvent μ b < x := by
by_cases hex : ∃ r, e < r ∧ measureResolvent μ r = x
· rw [measureInverse, dite_eq_left hex]
have hs := hex.choose_spec
constructor
· intro hrb
have ht := measureResolvent_strictAnti μ hμ hs.1 hrb
rwa [hs.2] at ht
· intro hR
by_contra hnot
rcases (le_of_not_gt hnot).eq_or_lt with h | h
· rw [h, hs.2] at hR
exact lt_irrefl _ hR
· have ht := measureResolvent_strictAnti μ hμ hb h
rw [hs.2] at ht
exact (not_lt_of_ge ht.le) hR
· rw [measureInverse, dite_eq_right hex]
have hv : measureResolvent μ b < x := by
by_contra hnot
exact hex (exists_measureResolvent_solution μ hμ hb hx (le_of_not_gt hnot))
exact ⟨fun _ => hv, fun _ => hb⟩
theorem continuousAt_measureInverse (μ : Measure ℝ) [IsProbabilityMeasure μ]
{e x : ℝ} (hμ : ∀ᵐ y ∂μ, y ≤ e) (hx : 0 < x) :
ContinuousAt (measureInverse μ e) x := by
have hbound := (measureInverse_bounds μ hμ hx).1
have hxp : ∀ᶠ y in 𝓝 x, 0 < y := Ioi_mem_nhds hx
apply tendsto_order.2
constructor
· intro l hl
by_cases hle : l < e
· filter_upwards [hxp] with y hy
exact hle.trans_le (measureInverse_bounds μ hμ hy).1
· let c := (l + measureInverse μ e x) / 2
have hec : e < c := by dsimp only [c]; linarith [le_of_not_gt hle]
have hlc : l < c := by dsimp only [c]; linarith
have hci : c < measureInverse μ e x := by dsimp only [c]; linarith
have hR : x < measureResolvent μ c := by
by_contra hnot
exact (not_le_of_gt hci)
((measureInverse_le_iff μ hμ hx hec).mpr (le_of_not_gt hnot))
filter_upwards [hxp, Iio_mem_nhds hR] with y hy hRy
have hnot : ¬ measureInverse μ e y ≤ c := by
intro hle
exact (not_le_of_gt hRy) ((measureInverse_le_iff μ hμ hy hec).mp hle)
exact hlc.trans (lt_of_not_ge hnot)
· intro u hu
have heu : e < u := hbound.trans_lt hu
have hR := (measureInverse_lt_iff μ hμ hx heu).mp hu
filter_upwards [hxp, Ioi_mem_nhds hR] with y hy hRy
exact (measureInverse_lt_iff μ hμ hy heu).mpr hRy
theorem continuousAt_measureR_pos (μ : Measure ℝ) [IsProbabilityMeasure μ]
{e x : ℝ} (hμ : ∀ᵐ y ∂μ, y ≤ e) (hx : 0 < x) :
ContinuousAt (measureR μ e) x := by
have hc : ContinuousAt (fun y => measureInverse μ e y - 1 / y) x :=
(continuousAt_measureInverse μ hμ hx).sub
(continuousAt_const.div continuousAt_id hx.ne')
apply hc.congr_of_eventuallyEq
filter_upwards [Ioi_mem_nhds hx] with y hy
simp only [measureR, ite_eq_left (show 0 < y from hy)]
end InvariantIsing
end
end OAI
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