Almost everywhere continuous functions

Let X be a locally compact σ compact Hausdorff space. Let µ be a complete regular Borel measure defined on the Borel sets of X. It is shown that there is a base for the topology of X consisting of open sets whose boundaries are of µ measure zero.

Let (S, p) be a metric space. It is shown that a function on X whose range is a subset of S can be uniformly approximated by µ almost everywhere continuous simple functions if, and only if, the function itself is µ almost everywhere continuous and its range is a totally bounded subset of S.

S is then specialized to be a Banach algebra and several consequences are obtained culminating in the study of the ideal structure of the ring of ail µ almost everywhere continuous functions on X whose ranges are totally bounded subsets of a Banach algebra which is either the reals, complexes or quaternions.