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b37e936 | 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 | 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
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