It is well known that a measurable homomorphism from a locally compact group G G to a topological group Y Y must be continuous if Y Y is either separable or σ \sigma -compact. In this work it is shown that the above requirement on Y Y can be somewhat relaxed and it is shown inter alia that a measurable homomorphism from a locally compact group to a locally compact abelian group will always be continuous. In addition, it is shown that if H H is a nonopen subgroup of a locally compact group, then under a variety of circumstances, some union of cosets of H H must fail to be measurable.