Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 3 / 22J / b / Solution

For a Borel set $A\subset [0,1)$, the two inverse branches of the doubling map give the disjoint setsEach has half the Lebesgue measure of $A$, by translation and scaling invariance. Hence $\boxed{\mu (T^{-1}A)=\mu (A)}$. The piecewise affine map is Borel measurable, so it is a measure-preserving transformation. The printed one-dimensional “baker's map” is this doubling map.