File size: 1,690 Bytes
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 | import OAI.Probability.FreeIsing.Energy.DiscrepancyRowEnergy
namespace OAI
noncomputable section
open MeasureTheory Filter Set
open scoped NNReal Topology
namespace Problem367.DiscrepancyEnergy
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
end Problem367.DiscrepancyEnergy
end
end OAI
|