Laczkovich proved that if bounded subsets $A$ and $B$ of $R^k$ have the same non-zero Lebesgue measure and the box dimension of the boundary of each set is less than $k$, then there is a partition of $A$ into finitely many parts that can be translate
d to form a partition of $B$. Here we show that it can be additionally required that each part is both Baire and Lebesgue measurable. As special cases, this gives measurable and translation-on
We give a sketch of proof that any two (Lebesgue) measurable subsets of the unit sphere in $R^n$, for $nge 3$, with non-empty interiors and of the same measure are equidecomposable using pieces that are measurable.
