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67aee1d | 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 | import OAI.Analysis.FreeEntropy.GaussianCorrection
namespace OAI
universe u1
noncomputable section
open MeasureTheory Set Filter
open scoped Topology ENNReal
namespace FiniteEntropySeparation
theorem integral_le_of_liminf_tests {Ω : Type u1} [MeasurableSpace Ω] (ν : Measure Ω)
(F : Ω → ℝ) (g : ℕ → Ω → ℝ) (hF : Integrable F ν) (hF0 : 0≤ᵐ[ν] F)
(hg : ∀k,Integrable (g k) ν) (hg0 : ∀k,0≤ᵐ[ν] g k)
(hl : ∀ᵐs ∂ν,ENNReal.ofReal (F s) ≤ liminf (fun k => ENNReal.ofReal (g k s)) atTop)
{C : ℝ} (hC : ∀ᶠk in atTop,(∫s,g k s ∂ν) ≤ C) : (∫s,F s ∂ν) ≤ C := by
have hc : 0≤C := by
obtain ⟨k,hk⟩ := hC.exists
exact (integral_nonneg_of_ae (hg0 k)).trans hk
apply (ENNReal.ofReal_le_ofReal_iff hc).mp
rw [ofReal_integral_eq_lintegral_ofReal hF hF0]
apply (lintegral_mono_ae hl).trans
apply (lintegral_liminf_le' (fun k => ENNReal.measurable_ofReal.comp_aemeasurable (hg k).aestronglyMeasurable.aemeasurable)).trans
apply liminf_le_of_frequently_le'
apply Eventually.frequently
filter_upwards [hC] with k hk
simp only [Function.comp_apply]
rw [← ofReal_integral_eq_lintegral_ofReal (hg k) (hg0 k)]
exact ENNReal.ofReal_le_ofReal hk
variable {n : ℕ} {D : ℕ → ℕ} (hD : ∀k,0<D k)
(μ : (k:ℕ) → Measure (MatrixCoordinates n (D k))) [∀k,IsProbabilityMeasure (μ k)]
(K : ℕ → ℝ) (hK : ∀k,0≤K k) (hμ : ∀k,∀ᵐa ∂μ k,‖a‖≤K k)
include hD hK hμ
theorem matrixFlow_limit_bound (F : ℝ → ℝ) {a t β L : ℝ}
(ha : 0<a) (hat : a≤t) (hF : IntervalIntegrable F volume a t)
(hF0 : ∀s∈Ioc a t,0≤F s)
(hl : ∀s∈Ioc a t,ENNReal.ofReal (F s) ≤ liminf (fun k => ENNReal.ofReal (normalizedMatrixFisher (μ k) s)) atTop)
(hinit : ∀k,β≤normalizedMatrixEntropy (μ k) a)
(hterm : ∀ε : ℝ,0<ε → ∀ᶠk in atTop,normalizedMatrixEntropy (μ k) t ≤ L+ε) :
β+(1/2:ℝ)*(∫s in a..t,F s) ≤ L := by
have ht : 0<t := ha.trans_le hat
apply le_of_forall_pos_le_add
intro ε hε
have hg k := normalizedMatrixFisher_intervalIntegrable (hD k) (μ k) (hK k) (hμ k) ha ht
have hbound : ∀ᶠk in atTop,(∫s in Ioc a t,normalizedMatrixFisher (μ k) s) ≤ 2*(L+ε-β) := by
filter_upwards [hterm ε hε] with k hk
have he := normalizedMatrixEntropy_flow (hD k) (μ k) (hK k) (hμ k) ha ht
rw [intervalIntegral.integral_of_le hat] at he
linarith [hinit k]
have hf := integral_le_of_liminf_tests (volume.restrict (Ioc a t)) F
(fun k => normalizedMatrixFisher (μ k)) hF.1
((ae_restrict_mem measurableSet_Ioc).mono (fun _ hs => hF0 _ hs))
(fun k => (hg k).1)
(fun k => (ae_restrict_mem measurableSet_Ioc).mono (fun s hs => normalizedMatrixFisher_nonneg (hD k) (μ k) (ha.trans hs.1)))
((ae_restrict_mem measurableSet_Ioc).mono (fun _ hs => hl _ hs)) hbound
rw [intervalIntegral.integral_of_le hat]
linarith
end FiniteEntropySeparation
end
end OAI
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