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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 | import OAI.Probability.InvariantIsing.Spectral.MeasureWeakResolvent
namespace OAI
/-! Spectral-transform continuity for compactly supported laws, including
the zero argument and the upper-edge branch. -/
noncomputable section
open MeasureTheory Filter Set
open scoped Topology Classical BoundedContinuousFunction
namespace InvariantIsing
theorem spectralMean_tendsto_weak
(μs : ℕ → ProbabilityMeasure ℝ) (μ : ProbabilityMeasure ℝ)
(hweak : Tendsto μs atTop (𝓝 μ)) {K : ℝ} (hK : 0 ≤ K)
(hs : ∀ k, ∀ᵐ y ∂(μs k : Measure ℝ), y ∈ Icc (-K) K)
(hμ : ∀ᵐ y ∂(μ : Measure ℝ), y ∈ Icc (-K) K) :
Tendsto (fun k => ∫ y, y ∂(μs k : Measure ℝ)) atTop
(𝓝 (∫ y, y ∂(μ : Measure ℝ))) := by
let f : ℝ → ℝ := fun y => max (-K) (min K y)
have hf : Continuous f := continuous_const.max (continuous_const.min continuous_id)
have hb y : |f y| ≤ K := by
rw [abs_le]
exact ⟨le_max_left _ _, max_le (by linarith) (min_le_left _ _)⟩
let F : ℝ →ᵇ ℝ := BoundedContinuousFunction.mkOfBound ⟨f, hf⟩ (2 * K) (by
intro y z
rw [Real.dist_eq]
exact (abs_sub _ _).trans ((add_le_add (hb y) (hb z)).trans_eq (by ring)))
have heq (ν : Measure ℝ) (hν : ∀ᵐ y ∂ν, y ∈ Icc (-K) K) :
(∫ y, F y ∂ν) = ∫ y, y ∂ν := by
apply integral_congr_ae
filter_upwards [hν] with y hy
change max (-K) (min K y) = y
rw [min_eq_right hy.2, max_eq_right hy.1]
have ht := (ProbabilityMeasure.tendsto_iff_forall_integral_tendsto.mp hweak) F
rw [heq (μ : Measure ℝ) hμ] at ht
exact ht.congr' (Eventually.of_forall fun k => heq (μs k : Measure ℝ) (hs k))
theorem measureR_tendsto_weak_pos
(μs : ℕ → ProbabilityMeasure ℝ) (μ : ProbabilityMeasure ℝ)
(es : ℕ → ℝ) (e : ℝ) (hweak : Tendsto μs atTop (𝓝 μ))
(hedge : Tendsto es atTop (𝓝 e))
(hs : ∀ k, ∀ᵐ y ∂(μs k : Measure ℝ), y ≤ es k)
(hμ : ∀ᵐ y ∂(μ : Measure ℝ), y ≤ e)
(xs : ℕ → ℝ) {x : ℝ} (hxs : Tendsto xs atTop (𝓝 x)) (hx : 0 < x) :
Tendsto (fun k => measureR (μs k : Measure ℝ) (es k) (xs k)) atTop
(𝓝 (measureR (μ : Measure ℝ) e x)) := by
have hi := measureInverse_tendsto_weak μs μ es e hweak hedge hs hμ xs hxs hx
have hrecip : Tendsto (fun k => 1 / xs k) atTop (𝓝 (1 / x)) :=
tendsto_const_nhds.div hxs hx.ne'
have ht : Tendsto
(fun k => measureInverse (μs k : Measure ℝ) (es k) (xs k) - 1 / xs k)
atTop (𝓝 (measureR (μ : Measure ℝ) e x)) := by
simpa only [measureR, ite_eq_left hx] using hi.sub hrecip
apply ht.congr'
filter_upwards [hxs.eventually (Ioi_mem_nhds hx)] with k hk
simp only [measureR, ite_eq_left (show 0 < xs k from hk)]
theorem measureR_tendsto_weak_zero
(μs : ℕ → ProbabilityMeasure ℝ) (μ : ProbabilityMeasure ℝ)
(es : ℕ → ℝ) (e : ℝ) (hweak : Tendsto μs atTop (𝓝 μ))
{K : ℝ} (hK : 0 < K) (he : e ≤ K) (hes : ∀ k, es k ≤ K)
(hs : ∀ k, ∀ᵐ y ∂(μs k : Measure ℝ), y ∈ Icc (-K) (es k))
(hμ : ∀ᵐ y ∂(μ : Measure ℝ), y ∈ Icc (-K) e)
(xs : ℕ → ℝ) (hxs : Tendsto xs atTop (𝓝 0)) :
Tendsto (fun k => measureR (μs k : Measure ℝ) (es k) (xs k)) atTop
(𝓝 (measureR (μ : Measure ℝ) e 0)) := by
have hm := spectralMean_tendsto_weak μs μ hweak hK.le
(fun k => (hs k).mono fun y hy => ⟨hy.1, hy.2.trans (hes k)⟩)
(hμ.mono fun y hy => ⟨hy.1, hy.2.trans he⟩)
have herr : ∀ᶠ k in atTop,
‖measureR (μs k : Measure ℝ) (es k) (xs k) - ∫ y, y ∂(μs k : Measure ℝ)‖ ≤
8 * K ^ 2 * |xs k| := by
filter_upwards [hxs.eventually (Iio_mem_nhds
(show (0 : ℝ) < 1 / (4 * K) by positivity))] with k hk
by_cases hxk : 0 < xs k
· have hsmall : 4 * K * xs k ≤ 1 := by
have := (lt_div_iff₀ (show (0 : ℝ) < 4 * K by positivity)).mp hk
nlinarith
have hb := measureR_sub_mean_bound (μs k : Measure ℝ) hK (hes k) (hs k) hxk hsmall
simpa only [Real.norm_eq_abs, abs_of_nonneg hb.1, abs_of_pos hxk] using hb.2
· simp only [measureR, ite_eq_right hxk, sub_self, norm_zero]
positivity
have hb : Tendsto (fun k => 8 * K ^ 2 * |xs k|) atTop (𝓝 0) := by
simpa using hxs.abs.const_mul (8 * K ^ 2)
have ht := (squeeze_zero_norm' herr hb).add hm
simpa only [sub_add_cancel, zero_add, measureR, lt_self_iff_false, ite_false] using ht
end InvariantIsing
end
end OAI
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