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A note on the Luzin-Menchoff theorem. (English) Zbl 1401.28001

A refinement of the Luzin-Menchoff theorem is given. It states that for every closed subset \(K\) of a measurable \(E\subset\mathbb R^n\) there is a closed set \(F\) such that \(K\subset F\subset E\), the upper densities of \(F\) and \(E\) at points of \(K\) coincide, the lower densities of \(F\) and \(E\) at points of \(K\) coincide, and the density of \(E\) at points of \(F\setminus K\) is one.

MSC:

28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
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