  We provide a concise, yet fairly complete discussion of the concept of
essential closures of subsets of the real axis and their intimate connection
with the topological support of absolutely continuous measures.
  As an elementary application of the notion of the essential closure of
subsets of $\bbR$ we revisit the fact that CMV, Jacobi, and Schr\"odinger
operators, reflectionless on a set E of positive Lebesgue measure, have
absolutely continuous spectrum on the essential closure of the set E (with
uniform multiplicity two on E). Though this result in the case of Schr\"odinger
and Jacobi operators is known to experts, we feel it nicely illustrates the
concept and usefulness of essential closures in the spectral theory of classes
of reflectionless differential and difference operators.
