File size: 2,899 Bytes
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 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 | import OAI.Geometry.RCD.Metric.L2IntegralRepresentative
namespace OAI
/-! # Pointwise representation of a bounded compact-support L2 integral -/
open Set Filter MeasureTheory
open scoped Topology ENNReal
noncomputable section
namespace RegularPointCoordinates
theorem bounded_support_L2_integral
{X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y]
(μ : Measure X) (ν : Measure Y) [SigmaFinite μ] [IsFiniteMeasure ν]
{s : Set X} (hs : MeasurableSet s) (hfin : μ s ≠ ∞)
(u : Y → Lp ℝ 2 μ) (hu : Integrable u ν) (f : Y → X → ℝ)
(hf : StronglyMeasurable (Function.uncurry f))
(hr : ∀ᵐ y ∂ν, u y =ᵐ[μ] f y) {A : ℝ}
(hb : ∀ᵐ y ∂ν, ∀ x, |f y x| ≤ A) (hz : ∀ᵐ y ∂ν, ∀ x, x ∉ s → f y x = 0) :
MemLp (fun x => ∫ y, f y x ∂ν) 2 μ ∧
(∫ y, u y ∂ν) =ᵐ[μ] fun x => ∫ y, f y x ∂ν := by
let g := fun x => ∫ y, f y x ∂ν
have hg0 (x : X) (hx : x ∉ s) : g x = 0 := by
exact integral_eq_zero_of_ae (hz.mono fun y hy => hy x hx)
have hgb (x : X) : ‖g x‖ ≤ A*ν.real univ :=
norm_integral_le_of_norm_le_const
(hb.mono fun y hy => by simpa only [Real.norm_eq_abs] using hy x)
have hgm : StronglyMeasurable g := hf.integral_prod_left
have hg : MemLp g 2 μ := by
let : IsFiniteMeasure (μ.restrict s) := ⟨by simpa using hfin.lt_top⟩
have hg' : MemLp g 2 (μ.restrict s) :=
MemLp.of_bound hgm.aestronglyMeasurable.restrict _ (Eventually.of_forall hgb)
have he : s.indicator g = g := by
funext x
by_cases hx : x ∈ s <;> simp [hx,hg0]
rw [← he]
exact (memLp_indicator_iff_restrict hs).mpr hg'
refine ⟨hg,L2_integral_representative μ ν u hu f hr ?_ g hg (EventuallyEq.refl _ _)⟩
intro ξ
let w := s.indicator (fun x => |ξ x|)
have hwi : Integrable w μ :=
(integrable_indicator_iff hs).mpr (lp_integrableOn_of_finite μ ξ s hfin).abs
have hbase : Integrable (fun q : Y × X => (1:ℝ)*w q.2) (ν.prod μ) :=
(integrable_const (1:ℝ)).mul_prod hwi
have hm : AEStronglyMeasurable (fun q : Y × X => ξ q.2*f q.1 q.2) (ν.prod μ) :=
((Lp.stronglyMeasurable ξ).comp_measurable measurable_snd).aestronglyMeasurable.mul
hf.aestronglyMeasurable
apply (hbase.const_mul A).mono' hm
have hpair : ∀ᵐ q ∂ν.prod μ, (∀ x, |f q.1 x| ≤ A) ∧
(∀ x, x ∉ s → f q.1 x = 0) :=
Measure.quasiMeasurePreserving_fst.tendsto_ae.eventually (hb.and hz)
filter_upwards [hpair] with q hq
rcases hq with ⟨hb,hz⟩
change ‖ξ q.2*f q.1 q.2‖ ≤ A*((1:ℝ)*w q.2)
by_cases hx : q.2 ∈ s
· simp only [Real.norm_eq_abs,abs_mul,w,indicator_of_mem hx,one_mul]
exact (mul_le_mul_of_nonneg_left (hb q.2) (abs_nonneg _)).trans_eq (mul_comm _ _)
· simp only [hz q.2 hx,mul_zero,norm_zero,w,indicator_of_notMem hx,le_refl]
end RegularPointCoordinates
end
end OAI
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