On the approximation of measurable functions by continuous functions
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We present a characterization of the completed Borel measure spaces for which every measurable function, with values in a separable Frechet space, is the almost everywhere limit of a sequence of continuous functions. From this characterization one can easily obtain results that have appeared recently in the literature, in a more general form. We also examine what happens when the range is a subset of an arbitrary Banach space, and show that this case often reduces to the separable case.
1980 Mathematics Subject Classification (1985 Revision)28C15 54D20
Key words and phrasesτ-smooth measures Lindelöf subsets
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