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f61e547 | 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 | import OAI.Probability.FreeIsing.Energy.NormalizedResponseLimit
namespace OAI
noncomputable section
open MeasureTheory Filter
open scoped Topology ENNReal
namespace Problem367
variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
theorem tsum_lintegral_sq_le_liminf_of_ae_tendsto
{ι : Type*} [Countable ι] (μ : Measure Ω)
{X : ℕ → ι → Ω → ℝ} {Y : ι → Ω → ℝ}
(hX : ∀ n i, AEStronglyMeasurable (X n i) μ)
(hlim : ∀ i, ∀ᵐ ω ∂μ, Tendsto (fun n => X n i ω) atTop (𝓝 (Y i ω))) :
(∑' i, ∫⁻ ω, ENNReal.ofReal ((Y i ω) ^ 2) ∂μ) ≤
liminf (fun n => ∑' i, ∫⁻ ω, ENNReal.ofReal ((X n i ω) ^ 2) ∂μ) atTop := by
have hcoord (i : ι) : (∫⁻ ω, ENNReal.ofReal ((Y i ω) ^ 2) ∂μ) ≤
liminf (fun n => ∫⁻ ω, ENNReal.ofReal ((X n i ω) ^ 2) ∂μ) atTop := by
calc
_ = ∫⁻ ω, liminf (fun n => ENNReal.ofReal ((X n i ω) ^ 2)) atTop ∂μ := by
apply lintegral_congr_ae
filter_upwards [hlim i] with ω hω
exact (ENNReal.tendsto_ofReal (hω.pow 2)).liminf_eq.symm
_ ≤ _ := lintegral_liminf_le' (fun n => ((hX n i).aemeasurable.pow_const 2).ennreal_ofReal)
apply (ENNReal.tsum_le_tsum hcoord).trans
let : MeasurableSpace ι := ⊤
have hmeas (n : ℕ) : Measurable
(fun i => ∫⁻ ω, ENNReal.ofReal ((X n i ω) ^ 2) ∂μ) := measurable_of_countable _
simpa only [lintegral_count] using
(lintegral_liminf_le (μ := Measure.count) hmeas)
theorem tsum_lintegral_sq_le_of_ae_tendsto
{ι : Type*} [Countable ι] (μ : Measure Ω)
{X : ℕ → ι → Ω → ℝ} {Y : ι → Ω → ℝ} {C : ℝ≥0∞}
(hX : ∀ n i, AEStronglyMeasurable (X n i) μ)
(hlim : ∀ i, ∀ᵐ ω ∂μ, Tendsto (fun n => X n i ω) atTop (𝓝 (Y i ω)))
(henergy : ∀ n, (∑' i, ∫⁻ ω, ENNReal.ofReal ((X n i ω) ^ 2) ∂μ) ≤ C) :
(∑' i, ∫⁻ ω, ENNReal.ofReal ((Y i ω) ^ 2) ∂μ) ≤ C :=
(tsum_lintegral_sq_le_liminf_of_ae_tendsto μ hX hlim).trans
(liminf_le_of_frequently_le' (Filter.Frequently.of_forall henergy))
theorem ae_summable_sq_of_tsum_lintegral_sq_ne_top
{ι : Type*} [Countable ι] (μ : Measure Ω) {Y : ι → Ω → ℝ}
(hY : ∀ i, AEStronglyMeasurable (Y i) μ)
(henergy : (∑' i, ∫⁻ ω, ENNReal.ofReal ((Y i ω) ^ 2) ∂μ) ≠ ⊤) :
∀ᵐ ω ∂μ, Summable (fun i => (Y i ω) ^ 2) := by
have hfinite : (∫⁻ ω, ∑' i, ENNReal.ofReal ((Y i ω) ^ 2) ∂μ) ≠ ⊤ := by
rw [lintegral_tsum (fun i => ((hY i).aemeasurable.pow_const 2).ennreal_ofReal)]
exact henergy
filter_upwards [ae_lt_top' (AEMeasurable.tsum
(fun i => ((hY i).aemeasurable.pow_const 2).ennreal_ofReal)) hfinite] with ω hω
have hs : Summable (fun i => ((Y i ω) ^ 2).toNNReal) :=
ENNReal.tsum_coe_ne_top_iff_summable.mp hω.ne
have hs' := NNReal.summable_coe.mpr hs
simpa only [Real.coe_toNNReal', max_eq_left (sq_nonneg _)] using hs'
theorem posteriorResponse_row_energy_le
{ι : Type*} [Countable ι] (μ : Measure Ω) [IsFiniteMeasure μ]
(F : Filtration ℕ mΩ) (f : Ω → ℝ) (g : ι → Ω → ℝ) {C : ℝ≥0∞}
(hinterior : ∀ᵐ ω ∂μ, μ[f | ⨆ n, F n] ω ∈ Set.Ioo (-1 : ℝ) 1)
(henergy : ∀ n, (∑' i, ∫⁻ ω,
ENNReal.ofReal ((posteriorResponse μ (F n) f (g i) ω) ^ 2) ∂μ) ≤ C) :
(∑' i, ∫⁻ ω,
ENNReal.ofReal ((posteriorResponse μ (⨆ n, F n) f (g i) ω) ^ 2) ∂μ) ≤ C := by
apply tsum_lintegral_sq_le_of_ae_tendsto μ
(fun n i => ((stronglyMeasurable_posteriorResponse μ (F n) f (g i)).mono
(F.le n)).aestronglyMeasurable)
(fun i => tendsto_ae_posteriorResponse μ F f (g i) hinterior) henergy
theorem integrable_tsum_sq_of_tsum_lintegral_sq_ne_top
{ι : Type*} [Countable ι] (μ : Measure Ω) {Y : ι → Ω → ℝ}
(hY : ∀ i, AEStronglyMeasurable (Y i) μ)
(henergy : (∑' i, ∫⁻ ω, ENNReal.ofReal ((Y i ω) ^ 2) ∂μ) ≠ ⊤) :
Integrable (fun ω => ∑' i, (Y i ω) ^ 2) μ := by
refine ⟨(AEMeasurable.tsum (fun i => (hY i).aemeasurable.pow_const 2)).aestronglyMeasurable, ?_⟩
have heq : (∫⁻ ω, ‖∑' i, (Y i ω) ^ 2‖ₑ ∂μ) =
∑' i, ∫⁻ ω, ENNReal.ofReal ((Y i ω) ^ 2) ∂μ := by
rw [← lintegral_tsum (fun i => ((hY i).aemeasurable.pow_const 2).ennreal_ofReal)]
apply lintegral_congr_ae
filter_upwards [ae_summable_sq_of_tsum_lintegral_sq_ne_top μ hY henergy] with ω hω
rw [Real.enorm_of_nonneg (tsum_nonneg (fun i => sq_nonneg _))]
exact ENNReal.ofReal_tsum_of_nonneg (fun i => sq_nonneg _) hω
change (∫⁻ ω, ‖∑' i, (Y i ω) ^ 2‖ₑ ∂μ) < ⊤
rw [heq]
exact henergy.lt_top
theorem integral_tsum_sq_le_of_tsum_lintegral_sq_le
{ι : Type*} [Countable ι] (μ : Measure Ω) {Y : ι → Ω → ℝ} {C : ℝ}
(hY : ∀ i, AEStronglyMeasurable (Y i) μ) (hC : 0 ≤ C)
(henergy : (∑' i, ∫⁻ ω, ENNReal.ofReal ((Y i ω) ^ 2) ∂μ) ≤ ENNReal.ofReal C) :
Integrable (fun ω => ∑' i, (Y i ω) ^ 2) μ ∧
(∫ ω, ∑' i, (Y i ω) ^ 2 ∂μ) ≤ C := by
have hfinite := ne_top_of_le_ne_top ENNReal.ofReal_ne_top henergy
have hint := integrable_tsum_sq_of_tsum_lintegral_sq_ne_top μ hY hfinite
refine ⟨hint, ?_⟩
have heq : ENNReal.ofReal (∫ ω, ∑' i, (Y i ω) ^ 2 ∂μ) =
∑' i, ∫⁻ ω, ENNReal.ofReal ((Y i ω) ^ 2) ∂μ := by
rw [ofReal_integral_eq_lintegral_ofReal hint
(Filter.Eventually.of_forall (fun ω => tsum_nonneg (fun i => sq_nonneg _)))]
rw [← lintegral_tsum (fun i => ((hY i).aemeasurable.pow_const 2).ennreal_ofReal)]
apply lintegral_congr_ae
filter_upwards [ae_summable_sq_of_tsum_lintegral_sq_ne_top μ hY hfinite] with ω hω
exact ENNReal.ofReal_tsum_of_nonneg (fun i => sq_nonneg _) hω
exact (ENNReal.ofReal_le_ofReal_iff hC).mp (heq.trans_le henergy)
end Problem367
end
end OAI
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