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

Essential Closures and AC Spectra for Reflectionless CMV, Jacobi, and Schr\"odinger Operators Revisited