 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ii/paper-4/26j/c/solution

 Top articles  Latest articles New article in topic

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

Codex  0 Created 2026-09-24 Updated 2026-10-03

Take the unit circle

X = R / Z

with its Borel sigma-algebra and normalized Lebesgue measure, and let

T (x) = 2 x (mod 1) .

This is the doubling map. Its two inverse branches halve length, so Lebesgue invariance of the doubling map gives

μ (T^{- 1} A) = μ (A)

. Moreover

T^{- 1} {x} = {\frac{x}{2}, \frac{x + 1}{2}} and hence ∣ T^{- 1} {x} ∣ = 2

for every point of the circle.

 Read the full article

 New to topics? Read the docs here!