## 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

### Keywords:

Luzin-Menchoff property; density; measurable set