I was studying Royden's construction of non measurable sets. There it is defined $x \equiv y$ iff $x-y \in \mathbb{Q}$ and $E$ to be set of all representatives of all conjugacy clases of $[0,1]$ under the equivalence relation. And is written $E+^{\circ}r$ is measurable if $E$ is where $E+^{\circ}r=\{e+^{\circ} r \mid e\in E\}$ where $x+^{\circ}y=x+y$ if $x+y<1$ and $x+y-1$ otherwise. Could one please explain me why $E+^{\circ}r$ is measurable if $E$ is ?
I've noticed if we can show $\mu^*(E+^{\circ} r)=\mu^*(E)$ then we're done. But here I'm stuck.
$E+^\circ r=((E\ \cap\ [0;1-r))+r)\ \cup \ ((E\ \cap\ [1-r;1])+r-1) $
Thus,
$E+^\circ r$ is the union of two measurable sets, since each set is the intersection of two measurable sets translated, and intersections of measurable sets are measurable as well as translations of measurable sets.
Thus, we are done.