= Uniform convergence on compacts in probability under an absolutely continuous measure change
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 $E$,
$$
Q(E)\leq K P(E)+\mathbb E_P[Z\mathbf1_{\{Z>K\}}].
$$
Integrability makes the second term arbitrarily small for large $K$. Thus $P(E_n)\to0$ implies $Q(E_n)\to0$; apply this to each event where the supremum on $[0,T]$ exceeds a fixed tolerance. Only <absolute continuity of measures> is needed, not equivalence.
Back to article page
