Exercise 8, § 4

If the torus $T = \mathbf{R}/\mathbf{Z}$ is canonically identified with the interval $[0, 1[$ of $\mathbf{R}$, the Lebesgue measure $\mu$ on this interval is a measure on $T$; for every set $A \subset T$ and every $z \in T$, $\mu^*(A + z) = \mu^*(A)$.

a) Show that there exists a subgroup $H_0$ of $T$ such that the sets $H_n = r_n + H_0$, where $r_n$ runs over the set of rational numbers contained in $[0, 1[$, form a partition of $T$ (regarding $\mathbf{R}$ as a vector space over $\mathbf{Q}$, note that in $\mathbf{R}$ the subspace $\mathbf{Q}$ admits a supplement).

b) Let $H$ be a set that is the union of any finite number of sets $H_n$; show that $H$ is not integrable and that $\mu_*(H) = 0$ (note that every subgroup of the additive group $\mathbf{Q}/\mathbf{Z}$ generated by a finite number of elements has infinite index in $\mathbf{Q}/\mathbf{Z}$; from this, deduce that there exists a partition of $T$ into a countable infinity of sets $P_n$, each of which contains a set of the form $z + H$).

c) Deduce from b) an example of a decreasing sequence $(A_n)$ of subsets of $T$, whose intersection is empty, and is such that $\mu^*(A_n) = 1$ for all $n$.

d) Let $A$ be an integrable set such that $H_0 \subset A$ and $\mu(A) = \mu^*(H_0) > 0$; show that, for every integrable set $B \subset A$, the sets $B_1 = B \cap H_0$ and $B_2 = B \cap \mathbf{C}H_0$ form a partition of $B$ such that $\mu^*(B_1) = \mu^*(B_2) = \mu(B)$ and $\mu_*(B_1) = \mu_*(B_2) = 0$ (make use of Exer. 7 b)).