Repeated <Cauchy-Schwarz inequality> shows that the $U^3$ norm dominates the absolute mean of a function:
$$
|\mathbb Ef|\leq\|f\|_{U^3}.
$$
Taking $f=1_A$ and using $\mathbb E1_A=\alpha$ gives
$$
\|1_A\|_{U^3}\geq\alpha.
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 129 / 4 / b / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-129/4/b/i/solution