| import OAI.Geometry.RCD.Metric.TimeAverageApproximation |
| import Mathlib.MeasureTheory.Integral.Prod |
|
|
| namespace OAI |
|
|
| /-! # Integrability of bounded time averages against a finite measure -/ |
|
|
| open Set Filter MeasureTheory |
| open scoped Topology |
| noncomputable section |
| namespace RegularPointCoordinates |
|
|
| theorem bounded_time_average_integrable |
| {Y : Type*} [MeasurableSpace Y] (ν : Measure Y) [IsFiniteMeasure ν] |
| (f : ℝ → Y → ℝ) (hf : StronglyMeasurable (Function.uncurry f)) |
| {h t M : ℝ} (hh : 0 < h) |
| (hb : ∀ᵐ y ∂ν, ∀ q ∈ Icc (t-h) (t+h), |f q y| ≤ M) : |
| Integrable (fun y => timeIntervalAverage (fun q => f q y) h t) ν := by |
| let τ := volume.restrict (Icc (t-h) (t+h)) |
| have hi : Integrable (Function.uncurry f) (τ.prod ν) := by |
| apply Integrable.of_bound hf.aestronglyMeasurable M |
| have hq : ∀ᵐ p ∂τ.prod ν, p.1 ∈ Icc (t-h) (t+h) := |
| Measure.quasiMeasurePreserving_fst.tendsto_ae.eventually (ae_restrict_mem measurableSet_Icc) |
| have hy : ∀ᵐ p ∂τ.prod ν, ∀ q ∈ Icc (t-h) (t+h), |f q p.2| ≤ M := |
| Measure.quasiMeasurePreserving_snd.tendsto_ae.eventually hb |
| filter_upwards [hq,hy] with p hp hbp |
| exact hbp p.1 hp |
| have ha := hi.integral_prod_right.const_mul ((2*h)⁻¹) |
| apply ha.congr |
| exact Eventually.of_forall fun y => by |
| change (2*h)⁻¹*(∫ q, f q y ∂τ) = (2*h)⁻¹*(∫ q in (t-h)..(t+h), f q y) |
| rw [intervalIntegral.integral_of_le (by linarith : t-h ≤ t+h), |
| ← integral_Icc_eq_integral_Ioc] |
|
|
| end RegularPointCoordinates |
|
|
| end |
|
|
| end OAI |
|
|
