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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 | import OAI.Probability.InvariantIsing.Spectral.MeasureSpectralContinuity
namespace OAI
/-! Uniform transform convergence and continuity of the variational value
for the compact spectral laws in manuscript Lemma `gen:R-continuity`. -/
noncomputable section
open MeasureTheory Filter Set
open scoped Topology Classical
namespace InvariantIsing
theorem measureR_tendsto_weak_nonneg
(μs : ℕ → ProbabilityMeasure ℝ) (μ : ProbabilityMeasure ℝ)
(es : ℕ → ℝ) (e : ℝ) (hweak : Tendsto μs atTop (𝓝 μ))
(hedge : Tendsto es atTop (𝓝 e))
{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) {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
rcases hx.eq_or_lt with hx | hx
· subst x
exact measureR_tendsto_weak_zero μs μ es e hweak hK he hes hs hμ xs hxs
· exact measureR_tendsto_weak_pos μs μ es e hweak hedge
(fun k => (hs k).mono fun _ h => h.2) (hμ.mono fun _ h => h.2) xs hxs hx
theorem measureR_uniform_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))
{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) :
∀ ε : ℝ, 0 < ε → ∀ᶠ k in atTop, ∀ x ∈ Icc (0 : ℝ) 1,
|measureR (μs k : Measure ℝ) (es k) x - measureR (μ : Measure ℝ) e x| < ε := by
intro ε hε
by_contra hbad
have hbad' : ∃ᶠ k in atTop, ∃ x ∈ Icc (0 : ℝ) 1,
ε ≤ |measureR (μs k : Measure ℝ) (es k) x - measureR (μ : Measure ℝ) e x| := by
simpa only [Filter.not_eventually, not_forall, not_imp, not_lt, exists_prop] using hbad
obtain ⟨φ, hφ, hφbad⟩ := extraction_of_frequently_atTop hbad'
choose xs hxs herr using hφbad
obtain ⟨x, hx, ψ, hψ, hlim⟩ := isCompact_Icc.tendsto_subseq hxs
have hR := measureR_tendsto_weak_nonneg (fun k => μs (φ (ψ k))) μ
(fun k => es (φ (ψ k))) e (hweak.comp (hφ.comp hψ).tendsto_atTop)
(hedge.comp (hφ.comp hψ).tendsto_atTop) hK he (fun k => hes (φ (ψ k)))
(fun k => hs (φ (ψ k))) hμ (fun k => xs (ψ k)) hlim hx.1
have hfixed := (continuous_measureR (μ : Measure ℝ) hK he hμ).continuousAt.tendsto.comp hlim
have hzero : Tendsto (fun k =>
|measureR (μs (φ (ψ k)) : Measure ℝ) (es (φ (ψ k))) (xs (ψ k)) -
measureR (μ : Measure ℝ) e (xs (ψ k))|) atTop (𝓝 0) := by
simpa only [Function.comp_apply, sub_self, abs_zero] using (hR.sub hfixed).abs
obtain ⟨k, hk⟩ := (hzero.eventually (Iio_mem_nhds hε)).exists
exact (not_le_of_gt hk) (herr (ψ k))
theorem measureVariational_ne_top_bot (μ : Measure ℝ) [IsProbabilityMeasure μ]
{K e : ℝ} (hK : 0 < K) (he : e ≤ K) (hμ : ∀ᵐ y ∂μ, y ∈ Icc (-K) e) :
variationalFunctional (measureR μ e) ≠ ⊤ ∧ variationalFunctional (measureR μ e) ≠ ⊥ := by
have hR := continuous_measureR μ hK he hμ
have hb x : |measureR μ e x - 0| ≤ K := by
have h := measureR_bounds_all μ hμ x
rw [sub_zero, abs_le]
exact ⟨h.1, h.2.trans he⟩
have hspec p : |spectralFunctional (measureR μ e) p| ≤ K / 2 := by
simpa only [spectralFunctional_constant, zero_div, sub_zero] using
abs_spectralFunctional_sub_le (measureR μ e) (fun _ => 0) hR continuous_const
K (fun x _ => hb x) p
have hupper : variationalFunctional (measureR μ e) ≤ (K / 2 : ℝ) := by
let p := constantPath 0 (by constructor <;> norm_num)
have hi := iInf_le (fun p : OverlapPath => entropyFunctional p +
(spectralFunctional (measureR μ e) p : EReal)) p
simp only [p, entropyFunctional_zero, zero_add] at hi
exact hi.trans (by exact_mod_cast (abs_le.mp (hspec p)).2)
have hlower : ((-(K / 2) : ℝ) : EReal) ≤ variationalFunctional (measureR μ e) := by
apply le_iInf
intro p
have hl : ((-(K / 2) : ℝ) : EReal) ≤ (spectralFunctional (measureR μ e) p : EReal) := by
exact_mod_cast (abs_le.mp (hspec p)).1
have hnonneg := add_le_add (entropyFunctional_nonneg p)
(le_rfl : (spectralFunctional (measureR μ e) p : EReal) ≤ _)
exact hl.trans (by simpa only [zero_add] using hnonneg)
exact ⟨ne_top_of_le_ne_top (EReal.coe_ne_top _) hupper,
ne_bot_of_le_ne_bot (EReal.coe_ne_bot _) hlower⟩
theorem measureVariational_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))
{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) :
Tendsto (fun k => (variationalFunctional (measureR (μs k : Measure ℝ) (es k))).toReal)
atTop (𝓝 (variationalFunctional (measureR (μ : Measure ℝ) e)).toReal) := by
have hf := measureVariational_ne_top_bot (μ : Measure ℝ) hK he hμ
have hR := continuous_measureR (μ : Measure ℝ) hK he hμ
apply Metric.tendsto_nhds.mpr
intro ε hε
filter_upwards [measureR_uniform_weak μs μ es e hweak hedge hK he hes hs hμ ε hε] with k hk
have hfk := measureVariational_ne_top_bot (μs k : Measure ℝ) hK (hes k) (hs k)
have hRk := continuous_measureR (μs k : Measure ℝ) hK (hes k) (hs k)
have hl := variationalFunctional_le_add _ _ hRk hR ε (fun x hx => (hk x hx).le)
have hr := variationalFunctional_le_add _ _ hR hRk ε
(fun x hx => by simpa only [abs_sub_comm] using (hk x hx).le)
rw [← EReal.coe_toReal hfk.1 hfk.2, ← EReal.coe_toReal hf.1 hf.2,
← EReal.coe_add, EReal.coe_le_coe_iff] at hl hr
rw [Real.dist_eq, abs_lt]
constructor <;> linarith
end InvariantIsing
end
end OAI
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