It is known that every finite
dimensional translation invariant subspace of the continuous functions on the real
line consists of exponential polynomials. The purpose of this paper is to prove
an analogous result under the hypotheses that the functions involved are
measurable instead of continuous (and two functions are considered identical if
they are equal almost everywhere) and that the functions are defined on a
σ-compact locally compact abelian group. There is an application of this
theorem to the characterization of differential operators at the end of the
paper.
