On the Lebesgue density theorem

The classical Lebesgue density theorem says that almost each point of a measurable set A is a density point of A . It is well known that the density point of a measurable set A can be described in terms of the convergence in measure of a sequence of characteristic functions of sets similar to A . In this note it is shown that in the Lebesgue density theorem the convergence in measure cannot be replaced by the convergence almost everywhere.