Maximal-minor integral differences for Lipschitz perturbations

The support lemma provides a compact containing and also for every sufficiently small positive . Lipschitz maps are locally integrable, so mollification is linear: . These mollifications are . Apply the smooth compact-perturbation theorem to and ; output selection preserves compact support. The chain rule identifies their derivatives with selected square operators and gives

Outside , is locally zero. The two maps then agree on a neighborhood, so their derivatives, and thus their selected minors, agree there. The previous integral therefore equals its restriction to . This argument only needs local equality; it does not require differentiability of the unsmoothed maps at every point.

Apply strong local convergence of mollified derivatives to and to , on this same compact . For , contractivity and linearity give

Continuous linear postcomposition preserves eventual membership; the limiting derivative belongs to by its Lipschitz bound. Thus all hypotheses of determinant-integral continuity hold with measure restricted to . Its output is for each of these two maps.

The smooth selected minors are integrable on compact . For the limiting maps , the bound and give the same integrability. Thus both terms may be integrated separately on . Using Step 3 for each term,

This proves the restricted difference integral is zero; it does not subtract unrestricted whole-space integrals.

Outside , the original perturbation is locally zero, so there as well. The difference is integrable on and equals zero outside; it is therefore integrable on the whole space and has the same integral. This proves the asserted whole-space identity without requiring mollification to preserve any boundary values.