  The property of Lebesgue measurability is familiar from analysis and here we shall be more brief. To begin with, one introduces a probability measure over sets of reals. A set of reals is null iff it has measure zero. A set of reals A is Lebesgue measurable iff it is “almost Borel” in the sense that there is a Borel set B such that the region where B and A do not overlap is null. The interest of this notion can perhaps be best brought out by recalling a theorem of Banach and Tarski. Banach and Tarski showed that it is possible to cut the unit sphere into finitely many pieces, and shift those pieces around (using only translations and rotations) to bring them back together into a sphere of twice the size. This requires a key application of the Axiom of Choice. It is immediate that such pieces cannot be Lebesgue measurable. The question is “How complex must such pieces be?” and this amounts to the question “Which sets are Lebesgue measurable?” It is clear that the Borel sets are Lebesgue measurable. Are the projective sets Lebesgue measurable? 