Vol. 21, No. 2, 1967

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An elementary proof that Haar measurable almost periodic functions are continuous

Henry Werner Davis

Vol. 21 (1967), No. 2, 241–248

It is known that a Haar measurable complex-valued (von Neumann) almost periodic function on a locally compact T0 topological group is continuous. For by applying the Bohr-von Neumann approximation theorem for almost periodic functions and the fact that a Haar measurable representation into the general linear group is necessarily continuous one may deduce that such a function is the uniform limit of a sequence of continuous functions. This approach, while straightforward, has the disativantage of depending on the very deep Bohrvon Neumann approximation theorem. The latter result is commonly proven through considerable usage of representation theory. This paper presents an alternative prøof that Haar measurability plus almost periodicity imply continuity. The proof is elementary in the sense that it uses only the basic definitions of almost periodic function theory and topology. It does, however, depend on the standard tools of measure theory.

Mathematical Subject Classification
Primary: 28.75
Secondary: 22.65
Received: 10 November 1965
Published: 1 May 1967
Henry Werner Davis