measure theory | Investigations in Forms

A point $p$ of a Lebesgue measurable set $A \subset \mathbb{R}^n$ is called the Lebesgue density point of $A$ if $\lim_{\epsilon \to 0} \frac{\mu(A \cap B(p,\epsilon))}{\mu(B(p,\epsilon))}=1$ where $B(p, \epsilon)$ denotes the open ball of radius $\epsilon$ about $p$ ; in some sense this means that $A$ takes 100% of the space about $p$ . In general, if the limit above exists (regardless of whether it’s equal to 1 or not), we call it’s value the Lebesgue density of $A$ at $p$ , denoted by $\mathrm{dens}_A(p)$ . Fixing any Lebesgue measurable set $A$ , there’s a theorem called Lebesgue density theorem which states that for almost every $x \in \mathbb{R}$ we have either $\mathrm{dens}_A(x)=1$ or $\mathrm{dens}_A(x)=0$ . The theorem looks reasonable if we think of simple shapes in space, for example if $A$ is a closed filled square in $\mathbb{R}^2$ , then the interior points have density $1$ while the points in the complement have density 0, and the points on the boundary have measure density $\frac{1}{2}$ , but the boundary of the square is a measure zero in $\mathbb{R}^2$ . Restricting our attention to $\mathbb{R}$ and thinking about it for the favorite set in real analysis, Cantor set, it’ll still make sense, because the Cantor set after all has measure zero itself, so those fractions are zero for all of the radii $\epsilon>0$ and point $x \in \mathbb{R}$ . Now one might wonder to twist the situation more by thinking about the fat Cantor set $C$ ; where the set now has some mass in contrast with the Cantor set. Then how do the density and non-density points (i.e. points where that limit is either not equal to 1 or it doesn’t exist) of $C$ look like? The fat Cantor set contains some easy to find Lebesgue non-density points: the boundary points of the open intervals that we remove at each step have density $\frac{1}{2}$ , let such points in $C$ be denoted by $\mathrm{Ends}(C)$ .