Von Neumann-Steinhaus decomposition of parallelepipeds

A problem of Steinhaus was to partition a finite interval $I$ of the real line into countably infinitely many pairwise disjoint sets that are congruent in the sense that each set is a translate of a fixed set $A$. This paper describes von Neumann’s solution of the Steinhaus problem and generalizes the result to parallelepipeds in ${\mathbf{R}}^n$.