Solution
= Solution
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.