Uniform convergence on compacts in probability (u.c.p.)

Processes $X_n$ converge uniformly on compacts in probability to $X$ when, for every $T<\infty$ and $\epsilon >0$,

If $Q\ll P$, uniform convergence on compacts in probability under $P$ implies the same convergence under $Q$. Indeed, let $Z=dQ/dP$ be the Radon-Nikodym derivative. For every event $E$,Integrability makes the second term arbitrarily small for large $K$. Thus $P(E_n)\to 0$ implies $Q(E_n)\to 0$; apply this to each event where the supremum on $[0,T]$ exceeds a fixed tolerance. Only absolute continuity of measures is needed, not equivalence.