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.
