Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 2 / 27L (Probability and Measure)

A measure is a map $\mu :\mathcal{A}\to [0,\infty ]$ with $\mu (\varnothing )=0$ and countable additivity on disjoint families.

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

The disjoint sets $E_k$ contain at most one applicable summand at each $x$. Thus $F_n$ stabilizes to $F=2^k$ when $x\in E_k$ and to $0$ when $f(x)=0$ or $\infty$. By definition $F\le f\le 2F$ (with the extended-value interpretation).

Monotone convergence gives $\int F={\sum }_k{2}^k\lambda (E_k)$. Since $F\le f\le 2F$, $f$ is integrable exactly when this series is finite.

The annulus where $2^k<|x{|}^{-b}\le 2^{k+1}$ has volume comparable to $2^{-kd/b}$ for large positive $k$. The series is therefore comparable to $\sum 2^{k(1-d/b)}$, which converges exactly when $b<d$.