One-selected-block Fubini closure for R-322 #
This file isolates the signed Fubini step used when the deterministic smallest-leftmost selector removes one proper primitive block. The first checkpoint is the exact product-measure reindex: selected coordinates are integrated on the inside and all complement coordinates remain on the outside.
theorem
Anderson4D.integral_pi_eq_integral_complement_integral_selected
{ι : Type u_1}
{X : Type u_2}
[Fintype ι]
[MeasurableSpace X]
(μ : MeasureTheory.Measure X)
[MeasureTheory.SigmaFinite μ]
(selected : ι → Prop)
[DecidablePred selected]
(f : (ι → X) → ℝ)
(hf : MeasureTheory.Integrable f (MeasureTheory.Measure.pi fun (x : ι) => μ))
:
(∫ (x : ι → X), f x ∂MeasureTheory.Measure.pi fun (x : ι) => μ) = ∫ (xC : { i : ι // ¬selected i } → X), ∫ (xB : { i : ι // selected i } → X), f
((MeasurableEquiv.piEquivPiSubtypeProd (fun (x : ι) => X) selected).symm
(xB, xC)) ∂MeasureTheory.Measure.pi fun (x : { i : ι // selected i }) =>
μ ∂MeasureTheory.Measure.pi fun (x : { i : ι // ¬selected i }) => μ
Exact signed Fubini reindex for one selected finite coordinate block.
The measurable equivalence remembers both pieces of the original tuple, so the integrand is not replaced by an absolute-value majorant. The selected coordinates occur in the inner integral, as required by the inside-to-outside collapse in paper Section 4.1.