| import OAI.Probability.FreeIsing.Energy.NormalizedResponseLimit |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory Filter |
| open scoped Topology ENNReal |
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| namespace Problem367 |
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| variable {Ω : Type*} [mΩ : MeasurableSpace Ω] |
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| 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) |
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| 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)) |
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| 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' |
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| 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 |
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| 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 |
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| 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) |
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| end Problem367 |
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| end |
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| end OAI |
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