On Lebesgue's Density Theorem

نویسنده

  • casper goffman
چکیده

The density theorem of Lebesgue [l] may be stated in the following form: If 5 is a measurable linear point set, the metric density of S exists and is equal to 0 or 1 almost everywhere. We prove the converse that for every set Z of measure 0 there is a measurable set 5 whose metric density does not exist at any point of Z. We note, however, that in order for Z to be the set of points for which the metric density of some set S exists but is different from 0 or 1, Z must be both of measure 0 and of first category. As a converse, we show that for every F, type set Z of measure 0, thus of first category, there is a measurable set S whose metric density exists but is different from 0 or 1 at every point of Z.

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تاریخ انتشار 2010