“Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them, without changing their original shape. However, the pieces themselves are not “solids” in the traditional sense, but infinite scatterings of points.” --- Wikipedia
It is a paradox because it contradicts basic geometric intuition: You should not be able construct two balls from the material in a single ball.
However, from a mathematical perspective, there is a clear explanation for this paradox: The key is in the word ‘material,’ there is not an infinite number of atoms in a bowling ball, but the canonical mathematical definition of a ball is of a convex space in which the space making up a ball is continuous. This, in turn, sets the trap of measure theory, as assigning a non-zero measure to every subspace will lead to contradictions like this. This is the foundation to why we need sigma fields over which to define our probabilities.