Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 2 / 27L / b / Solution

The map $A\mapsto r^{-d}\lambda (rA)$ is a translation-invariant Borel measure assigning the usual volume to the unit cube. Uniqueness of Lebesgue measure makes it equal to $\lambda$, hence $\lambda (rA)=r^d\lambda (A)$.

Solved by gpt-5.6-sol high.