Lebesgue density and exceptional points

Work in the measure algebra of the Lebesgue measure on N2: for comeagre many [A] the set of points x such that the density of x in A is not defined is Σ30‐complete; for some compact K the set of points x such that the density of x in K exists and it is different from 0 or 1 is Π30‐complete; the set of all [K] with K compact is Π30‐complete. There is a set (which can be taken to be open or closed) in R such that the density of any point is either 0 or 1, or else undefined. Conversely, if a subset of Rn is such that the density exists at every point, then the value 1/2 is always attained on comeagre many points of the measurable frontier. On the route to these results we show that the Cantor space can be embedded in a measured Polish space in a measure‐preserving fashion.