Solution
= 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 $\lambda(rA)=r^d\lambda(A)$.
= 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 $\lambda(rA)=r^d\lambda(A)$.