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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 | import OAI.Probability.InvariantIsing.Spectral.MeasureRBounds
namespace OAI
/-! Resolvent continuity under weak convergence and convergence of upper
edges. Bounded continuous test functions avoid the pole outside the supports. -/
noncomputable section
open MeasureTheory Filter Set
open scoped Topology Classical BoundedContinuousFunction
namespace InvariantIsing
theorem measureResolvent_tendsto_weak
(μs : ℕ → ProbabilityMeasure ℝ) (μ : ProbabilityMeasure ℝ)
(es : ℕ → ℝ) (e : ℝ) (hweak : Tendsto μs atTop (𝓝 μ))
(hedge : Tendsto es atTop (𝓝 e))
(hsupport : ∀ k, ∀ᵐ y ∂(μs k : Measure ℝ), y ≤ es k)
(hsupport0 : ∀ᵐ y ∂(μ : Measure ℝ), y ≤ e)
{b : ℝ} (hb : e < b) :
Tendsto (fun k => measureResolvent (μs k : Measure ℝ) b) atTop
(𝓝 (measureResolvent (μ : Measure ℝ) b)) := by
let c := (e + b) / 2
let d := b - c
have hec : e < c := by dsimp only [c]; linarith
have hcb : c < b := by dsimp only [c]; linarith
have hd : 0 < d := sub_pos.mpr hcb
let f : ℝ → ℝ := fun y => 1 / max d (b - y)
have hf : Continuous f := continuous_const.div
(continuous_const.max (continuous_const.sub continuous_id))
(fun y => ne_of_gt (hd.trans_le (le_max_left _ _)))
have hfb y : |f y| ≤ 1 / d := by
dsimp only [f]
rw [abs_of_pos (one_div_pos.mpr (hd.trans_le (le_max_left _ _)))]
exact one_div_le_one_div_of_le hd (le_max_left _ _)
let F : ℝ →ᵇ ℝ := BoundedContinuousFunction.mkOfBound ⟨f, hf⟩ (2 / d) (by
intro y z
rw [Real.dist_eq]
calc
|f y - f z| ≤ |f y| + |f z| := abs_sub _ _
_ ≤ 1 / d + 1 / d := add_le_add (hfb y) (hfb z)
_ = 2 / d := by ring)
have heq (ν : Measure ℝ) (hν : ∀ᵐ y ∂ν, y ≤ c) :
(∫ y, F y ∂ν) = measureResolvent ν b := by
apply integral_congr_ae
filter_upwards [hν] with y hy
change 1 / max d (b - y) = 1 / (b - y)
rw [max_eq_right (by dsimp only [d]; linarith)]
have ht := (ProbabilityMeasure.tendsto_iff_forall_integral_tendsto.mp hweak) F
rw [heq (μ : Measure ℝ) (hsupport0.mono fun y hy => hy.trans hec.le)] at ht
apply ht.congr'
filter_upwards [hedge.eventually (Iio_mem_nhds hec)] with k hk
exact heq (μs k : Measure ℝ) ((hsupport k).mono fun y hy => hy.trans hk.le)
theorem measureInverse_tendsto_weak
(μs : ℕ → ProbabilityMeasure ℝ) (μ : ProbabilityMeasure ℝ)
(es : ℕ → ℝ) (e : ℝ) (hweak : Tendsto μs atTop (𝓝 μ))
(hedge : Tendsto es atTop (𝓝 e))
(hsupport : ∀ k, ∀ᵐ y ∂(μs k : Measure ℝ), y ≤ es k)
(hsupport0 : ∀ᵐ y ∂(μ : Measure ℝ), y ≤ e)
(xs : ℕ → ℝ) {x : ℝ} (hxs : Tendsto xs atTop (𝓝 x)) (hx : 0 < x) :
Tendsto (fun k => measureInverse (μs k : Measure ℝ) (es k) (xs k)) atTop
(𝓝 (measureInverse (μ : Measure ℝ) e x)) := by
have hbound := (measureInverse_bounds (μ : Measure ℝ) hsupport0 hx).1
have hxp : ∀ᶠ k in atTop, 0 < xs k := hxs.eventually (Ioi_mem_nhds hx)
apply tendsto_order.2
constructor
· intro l hl
by_cases hle : l < e
· filter_upwards [hxp, hedge.eventually (Ioi_mem_nhds hle)] with k hk hek
exact hek.trans_le (measureInverse_bounds (μs k : Measure ℝ) (hsupport k) hk).1
· let c := (l + measureInverse (μ : Measure ℝ) 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 : Measure ℝ) e x := by dsimp only [c]; linarith
have hR : x < measureResolvent (μ : Measure ℝ) c := by
by_contra hnot
exact (not_le_of_gt hci)
((measureInverse_le_iff (μ : Measure ℝ) hsupport0 hx hec).mpr (le_of_not_gt hnot))
have hRc := measureResolvent_tendsto_weak μs μ es e hweak hedge hsupport hsupport0 hec
filter_upwards [hxp, hedge.eventually (Iio_mem_nhds hec), hxs.eventually_lt hRc hR]
with k hk hek hRk
apply hlc.trans
apply lt_of_not_ge
intro hnot
exact (not_le_of_gt hRk)
((measureInverse_le_iff (μs k : Measure ℝ) (hsupport k) hk hek).mp hnot)
· intro u hu
have heu : e < u := hbound.trans_lt hu
have hR := (measureInverse_lt_iff (μ : Measure ℝ) hsupport0 hx heu).mp hu
have hRu := measureResolvent_tendsto_weak μs μ es e hweak hedge hsupport hsupport0 heu
filter_upwards [hxp, hedge.eventually (Iio_mem_nhds heu), hRu.eventually_lt hxs hR]
with k hk hek hRk
exact (measureInverse_lt_iff (μs k : Measure ℝ) (hsupport k) hk hek).mpr hRk
end InvariantIsing
end
end OAI
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