Vol. 19, No. 2, 1966

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Some lower bounds for Lebesgue area

William P. Ziemer

Vol. 19 (1966), No. 2, 381–390

It is well known in area theory that a continuous map f of the unit square Q2 into Euclidean space E2 can have zero Lebesgue area even though its range has a nonempty interior. This cannot happen if f is suitably well-behaved, for example, if f is light, Lipschitzian, or as we shall see below, if f satisfies a certain interiority condition. The purpose of this paper is to determine conditions under which an arbitrary measurable set A Q2 will support the Lebesgue area of f. The results imply that if fA is Lipschitz and if one of the coordinate functions of f is BV T (and continuous), then the Lebesgue area of f is no less than the integral of the multiplicity function N(f,A,y), where N(f,A,y) is the number (possibly ) of points in f1(y) A. We show that the BV T condition cannot be omitted. The proofs of theorems involving Lebesgue area depend upon a new co-area formula for real valued BV T functions.

Mathematical Subject Classification
Primary: 28.80
Received: 30 August 1965
Published: 1 November 1966
William P. Ziemer