Solution
= Solution
On disjoint events $A_n$ with $P(A_n)=c/n^2$, set $X_n=n1_{A_n}$. Then $X_n\to0$ almost surely and $E X_n^2=c$, but any common dominator has <integral> at least $\sum_n nP(A_n)=c\sum1/n=\infty$. Thus no integrable dominating variable exists.