This is Problem 9 of Chapter 1 of Stein and Shakarchi's Functional Analysis.
As a consequence of the previous problem one can show that it is not possible to extend Lebesgue measure on $\mathbb{R}^d$ , $d \ge 3$, as a finitely-additive measure on all subsets of $\mathbb{R}^d$ so that it is both translation and rotation invariant (that is, invariant under Euclidean motions). This is graphically shown by the “Banach-Tarski paradox”: There is a finite decomposition of the unit ball $B_1 = \sum_{j=1}^N E_j$, with the sets $E_j$ disjoint, and there are corresponding sets $\tilde{E}_j$ that are each obtained from $E_j$ by a Euclidean motion, with the $\tilde{E}_j$ also disjoint, so that $\sum_{j=1}^N \tilde{E}_j = B_2$ the ball of radius 2.
The "previous problem" refers to the Hausdorff Paradox.
The Banach-Tarski paradox, as far as I know, is a proof the a unit ball can be doubled into two unit balls through rotations of its component sets. But this problem is about doubling the radius of the unit ball through rotations and translations.