Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 210 / 1 / c

No. For each realized sample let $A_n=\{X_1,\dots ,X_n\}$. This is a finite Borel set, and continuity of $F$ makes $P(A_n)=0$ almost surely, while $P_n(A_n)=1$. Thereforealmost surely for every $n$, so the class of all Borel-set indicators cannot satisfy a ULLN.