We come up with a simple proof for a continuous version of the Hausdorff–Banach– Tarski paradox, which does not make use of Robinson’s method of compatible congruences and fits in the case of finite and countable paradoxical decompositions. It is proved that there exists a free subgroup whose rank is of the power of the continuum in a rotation group of a three-dimensional Euclidean space. We also argue that unbounded subsets of Euclidean space containing inner points are denumerably equipollent. Hausdorff [1, pp. 5-10] proved that there is no finitely additive measure that is defined on all subsets of a Euclidean space R 3 and is invariant under space motions. The main reason is the existence of nontrivial free products and free non-Abelian subgroups in a space rotation group. Banach and Tarski [2], developing Hausdorff’s construction, demonstrated that every two bounded sets containing inner points are equipollent; that is, such sets can be decomposed into a finite and equal number of parts so that the parts of the two sets are mutually isometric under some mapping. In other words, a first set may be divided into a finite number of parts in a way that after the parts are moved in space and joined together we obtain a second set. In particular, any sphere and polyhedron are equipollent. If their measures are different, then all parts involved in partitioning are not measurable. This is properly the well-known Hausdorff–Banach–Tarski paradox. Robinson [3] minimized the number of parts in a paradoxical decomposition of a sphere using the method of compatible congruencies. That method was widely used in subsequent research—in particular, in cases where continuous decompositions were allowed, but along with finite assemblies only. Mycielski [4], using Robinson’s method and Sierpinski’s construction [5] of a free subgroup of power-of-the-continuum rank in a rotation group of a three-dimensional Euclidean space, derived ∗ Supported by the Grants Council (under RF President) for State Aid of Leading Scientific Schools, grant