Vol. 20, No. 1, 1967

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A theorem on lattice ordered groups, results of Ptak, Namioka and Banach, and a front-ended proof of Lebesgue’s theorem

Stephen Simons

Vol. 20 (1967), No. 1, 149–153

The main theorem in this paper is on (not necessarily commutative) lattice ordered groups, and is a generalization of a result on finitely additive set functions due to Namioka. Our result can be used to prove Ptak’s combinatorial theorem on convex means, to give a short non measure-theoretic proof of Lebesgue’s dominated convergence theorem for a sequence of continuous functions on a countably compact topological space, and to give a short proof of Banach’s criteria for the weak convergence of a sequence in the Banach space of all bounded, real functions on an abstract set.

Mathematical Subject Classification
Primary: 46.06
Received: 19 November 1965
Published: 1 January 1967
Stephen Simons