Haar-product integrability for one lifted Green edge #
This deterministic helper adjoins one translated Green edge to an integrable
function on an arbitrary SFinite measure space. It is the common induction
step for both deterministic increasing Green trees and random-parametrix
Green paths, so it lives below both layers.
theorem
Anderson4D.integrable_greenFn_sub_mul_lift
{Y : Type u_1}
[MeasurableSpace Y]
{ν : MeasureTheory.Measure Y}
[MeasureTheory.SFinite ν]
(f : Y → ℂ)
(shift : Y → T4)
(hf : MeasureTheory.Integrable f ν)
(hshift : Measurable shift)
:
MeasureTheory.Integrable (fun (p : T4 × Y) => ↑(greenFn (p.1 - shift p.2)) * f p.2) (paperMeasure.prod ν)
Multiplying an integrable function by a translated Green kernel in a new Haar variable preserves joint integrability. The translation may depend measurably on all old variables.