Exercise 7, Appendix 0

a) Let $\mathfrak{B}$ be the set of bounded subsets of the plane $\mathbf{R}^2$ and let $\Gamma$ be the group of displacements of $\mathbf{R}^2$; let $C$ be the square $[0, 1] \times [0, 1]$. Show that there exists a positive additive set function $\lambda$ defined on $\mathfrak{B}$, which is invariant under $\Gamma$ and is such that $\lambda(C) = 1$ (apply exerc. 6 to the case where $g$ is the characteristic function of $C$; observe that $\Gamma$ is solvable, hence has an invariant mean (IV, p. 41, corollary)). If $A$ is a bounded subset of $\mathbf{R}^2$, whose boundary is negligible for the Lebesgue measure $\mu$ on $\mathbf{R}^2$, then $\lambda(A) = \mu(A)$ (for every $\varepsilon > 0$, there exists two sets $A_1$ and $A_2$ such that $A_1 \subset A \subset A_2$, $\mu(A_2 - A_1) < \varepsilon$, and $A_1$ and $A_2$ are unions of a finite number of squares). b) Consider the same question as in a), taking for $\mathfrak{B}$ the set of all subsets of $\mathbf{R}^2$, for $\Gamma$ the group of similarity transformations and $C = \mathbf{R}^2$.