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 f∣A 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 f−1(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.
