Abstract
Given a real-valued upper semicontinuous function h and a real-valued lower semicontinuous function g on a metric space such that (1) h ≥ g pointwise and (2) h(x) = g(x) at each isolated point of the space, it is not in general possible to find a real-valued function f whose upper envelope is h and whose lower envelope is g, even if the space is compact and dense-in-itself. The purpose of this note is to show that such an f exists in the case that both h and g are continuous, and that f can be chosen to be a Borel function.
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To Michel Thera, on his 70th birthday.
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Beer, G. Functions with Continuous Upper and Lower Envelopes. Vietnam J. Math. 46, 169–175 (2018). https://doi.org/10.1007/s10013-017-0266-7
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DOI: https://doi.org/10.1007/s10013-017-0266-7