Exercise 15, § 6

Denoting by $\mu$ the Lebesgue measure on $\mathbf{R}$, a point $x \in \mathbf{R}$ is said to be a density point of a subset $A$ of $\mathbf{R}$ if, when $h$ and $k$ tend to 0 through values $> 0$, the quotient $\mu^*(A \cap [x-h, x+k])/(h+k)$ tends to 1. Show that the set of points of an arbitrary subset $A$ of $\mathbf{R}$ that are not density points of $A$ is negligible for $\mu$. (Reduce to the case that $A$ is contained in a compact interval $[a, b]$, and consider the function $s_A(x) = \mu^*(A \cap [a, x])$. Take a decreasing sequence $(A_n)$ of open sets containing $A$, such that the series with general term $s_{A_n}(x) - s_A(x)$ is convergent on $[a, b]$, and use Exer. 13.)