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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 ℝ we revisit the fact that CMV, Jacobi, and Schrödinger operators, reflectionless on a set ℰ of positive Lebesgue...
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