| import OAI.Probability.FreeIsing.Energy.DiscrepancyRowEnergy |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped NNReal Topology |
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| namespace Problem367.DiscrepancyEnergy |
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| theorem hasSum_intervalIntegral_sq_row |
| {Ω ι : Type*} [MeasurableSpace Ω] [Countable ι] |
| {μ : Measure Ω} [SFinite μ] {c : ι → ℝ≥0 → Ω → ℝ} |
| (T : ℝ≥0) |
| (hc : ∀ i, AEStronglyMeasurable |
| (fun p : ℝ × Ω => c i p.1.toNNReal p.2) |
| ((volume.restrict (Ioc (0 : ℝ) T)).prod μ)) |
| (hs : ∀ᵐ p ∂(volume.restrict (Ioc (0 : ℝ) T)).prod μ, |
| Summable (fun i => c i p.1.toNNReal p.2 ^ 2)) |
| (hA : Integrable (fun p : ℝ × Ω => ∑' i, c i p.1.toNNReal p.2 ^ 2) |
| ((volume.restrict (Ioc (0 : ℝ) T)).prod μ)) : |
| HasSum (fun i => ∫ t in (0 : ℝ)..(T : ℝ), |
| ∫ ω, c i t.toNNReal ω ^ 2 ∂μ) |
| (∫ p : ℝ × Ω, ∑' i, c i p.1.toNNReal p.2 ^ 2 |
| ∂(volume.restrict (Ioc (0 : ℝ) T)).prod μ) := by |
| have hi (i : ι) : Integrable (fun p : ℝ × Ω => c i p.1.toNNReal p.2 ^ 2) |
| ((volume.restrict (Ioc (0 : ℝ) T)).prod μ) := by |
| apply hA.mono' ((hc i).pow 2) |
| filter_upwards [hs] with p hp |
| rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)] |
| exact hp.le_tsum i (fun j _ => sq_nonneg _) |
| have he (i : ι) : |
| (∫ p : ℝ × Ω, c i p.1.toNNReal p.2 ^ 2 |
| ∂(volume.restrict (Ioc (0 : ℝ) T)).prod μ) = |
| ∫ t in (0 : ℝ)..(T : ℝ), ∫ ω, c i t.toNNReal ω ^ 2 ∂μ := by |
| rw [integral_prod _ (hi i), intervalIntegral.integral_of_le T.coe_nonneg] |
| simpa only [he] using hasSum_integral_sq_row hc hs hA |
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| end Problem367.DiscrepancyEnergy |
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| end |
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| end OAI |
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