Exercise 24, § 1

a) In $\mathbf{R}$, let $A$ (resp. $B$) be the set of numbers $x$ whose proper dyadic expansion $x = x_0 + \sum_{i=1}^\infty x_i 2^{-i}$ ($x_0$ integer, $x_i = 0$ or $x_i = 1$ for $i \geq 1$) is such that $x_i = 0$ for $i$ even $> 0$ (resp. $i$ odd). Prove that, for Lebesgue measure, $A$ and $B$ are of 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 the numbers $rh$, where $r \in \mathbf{Q}$ and $h \in H$, is also of measure zero.

c) Let $P_n$ denote the set of real numbers for which at most $n$ coordinates relative to the basis $H$ are non-zero. Prove that if $P_n$ is negligible and if $P_{n+1}$ is measurable, then $P_{n+1}$ is negligible. (Let $h_0 \in H$; first prove that the set $S$ of the $x \in P_{n+1}$ whose coordinate relative to $h_0$ is $\neq 0$, is negligible. Using exerc. 20, prove 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 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.