Your quoted theorem (actually, a generalization where the second set is $[0,n)$ or even $[0,\infty)$) appears as Lemme 32, p. 267 of the [Banach-Tarski paper (1924)](http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-fmv6i1p27bwm).