Slices, multiplicity, and Lebesgue area

Wilfred Dennis Pepe and William P. Ziemer

Vol. 43 (1972), No. 3, 701–710

DOI: 10.2140/pjm.1972.43.701

Abstract

For a large class of k dimensional surfaces, S, it is shown that the Lebesgue area of S can be approximated by the integral of the k − 1 area of a family, F, of k − 1 dimensional surfaces that cover S. The family F is regarded as being composed of the slices of the surface S. In addition, a topological characterization of a certain multiplicity function is given. This multiplicity function when integrated with respect to k dimensional Hausdorff measure, yields the Lebesgue area of f.

Mathematical Subject Classification 2000

Primary: 28A75

Secondary: 49F25

Milestones

Received: 25 August 1971

Published: 1 December 1972