Some lower bounds for Lebesgue area
William P. Ziemer · Pacific Journal of Mathematics · 1966
It is well known in area theory that a continuous map / of the unit square Q 2 into Euclidean space E 2 can have zero Lebesgue area even though its range has a nonempty interior.This cannot happen if / is suitably well-behaved, for example, if / is light, Lipschitzian, or as we shall see below, if / satisfies a certain interiority condition.The purpose of this paper is to determine conditions under which an arbitrary measurable set AcQ 2 will support the Lebesgue area of /.The results imply that if /1 A is Lipschitz and if one of the coordinate functions of / is BVT (and continuous), then the Lebesgue area of / is no less than the integral of the multiplicity function N(f, A, y), where N(f, A, y) is the number (possibly oo) of points in f~Ky) Π A. We show that the BVT condition cannot be omitted.The proofs of theorems involving Lebesgue area depend upon a new co-area formula for real valued BVT functions.2* Preliminaries* Our proofs rely heavily upon the following