Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 4 / 26J / c / Solution

Take the unit circle $X=\mathbb{R}/\mathbb{Z}$ with its Borel sigma-algebra and normalized Lebesgue measure, and letThis is the doubling map. Its two inverse branches halve length, so Lebesgue invariance of the doubling map gives $\mu (T^{-1}A)=\mu (A)$. Moreoverfor every point of the circle.