Exercise 24, § 1

a) In $\mathbf{R}$, let $A$ (resp. $B$) be the set of numbers $x$ whose proper dyadic expansion $x = x_0 + \sum_{i=1}^\infty x_i 2^{-i}$ ($x_0$ an integer, $x_i = 0$ or $x_i = 1$ for $i \geq 1$) is such that $x_i = 0$ for $i$ even and $> 0$ (resp. $i$ odd). Show that, for Lebesgue measure, $A$ and $B$ have measure zero but $A + B = \mathbf{R}$.

b) Deduce from a) that there exists a basis $H$ of $\mathbf{R}$ (over $\mathbf{Q}$) contained in $A \cup B$, hence of measure zero. The set $P_1$ of numbers $rh$, where $r \in \mathbf{Q}$ and $h \in H$, is also of measure zero.

c) Denote by $P_n$ the set of real numbers at most $n$ of whose coordinates relative to the basis $H$ are nonzero. Show that if $P_n$ is negligible and $P_{n+1}$ is measurable, then $P_{n+1}$ is negligible. (Let $h_0 \in H$; show first that the set $S$ of $x \in P_{n+1}$ whose coordinate relative to $h_0$ is $\neq 0$, is negligible. Using Exer. 20, show that if $P_{n+1}$ were not negligible, there would exist two distinct points $x', x''$ of $P_{n+1} \cap \mathbf{C}S$ such that $(x' - x'')/h_0$ is rational, and deduce from this a contradiction.)

d) Deduce from a) and b) that there exist in $\mathbf{R}$ two negligible sets $C, D$ such that $C + D$ is not measurable.