a1 The School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW.
We denote by S the unit sphere in 3, and µ is the rotationally invariant measure, generalizing surface area on S; thus µS = 4π. We identify directions (or unit vectors) in 3 with points on S, and prove the following:
Theorem 1. If E is a subset of 3 of Lebesgue measure zero, then for µ almost all directions α, every plane normal to α intersects E in a set of plane measure zero.
(Received June 06 1979)