  Work in the measure algebra of the Lebesgue measure on the Cantor space: for
comeager many $[A]$ the set of points $x$ such that the density of $x $ at $A$
is not defined is $\Sigma^{0}_{3}$-complete; for some compact $K$ the set of
points $x$ such that the density of $x$ at $K$ exists and it is different from
$0$ or $1$ is $\Pi^{0}_{3}$-complete; the set of all $[K]$ with $K$ compact is
$\Pi^{0}_{3}$-complete. There is a set (which can be taken to be open or
closed) in $\mathbb R$ such that the density of any point is either $0$ or $1$,
or else undefined. Conversely, if a subset of $\mathbb R^n$ is such that the
density exists at every point, then the value $1/2$ is always attained. On the
route to this result we show that Cantor space can be embedded in a measured
Polish space in a measure-preserving fashion.
