J Austral Math Soc Ser A 45 pp83--94, 1988.

Some Inequalities that Arise in Measure Theory

Gavin Brown

Abstract

The Lesbesgue measure, l(E + F), of the algebraic sum of two Borel sets, E, F of the classical "middle-thirds" Cantor set on the circle can be estimated by evaluating the Cantor measure, m, of the summands. For example log l(E + F) exceeds a fixed scalar multiple of log m(E) + log m(F). Several numerical inequalities which are required to prove this and related results look tantalizingly simple and basic. Here we isolate them from the measure theory and present a common format and proof.

1980 AMS Subject Classification: primary 26D20; secondary 43A05, 43A10

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Authors

Gavin Brown

School of Mathematics, University of New South Wales, Kensington, N.S.W 2033, Australia.