I am looking for the original author and the date of publication of the following result.

Theorem There exist subsets $E_i\subset [0,1)$, $i\in {\bf Z}$, pairwise disjoints and real numbers $a_i$ such that $$ [0,1) = \coprod_{i\in {\bf Z}} E_i, \quad [0,2) = \coprod_{i\in {\bf Z}} a_i + E_i. $$ I am pretty sure that this is either Borel or Lebesgue. I remember having seen the result in the collected works of one of these two mathematicians, who advertised it as a reason to reject the axiom of choice.