Convergence in equivalent probability measure

Question

Let $(\Omega, \mathcal{F}, P)$ be a probability space and $A_n$ be a sequence of events such that $P(A_n)$ converge to 0. If $Q$ is an equivalent probability measure of $P$, does it mean that $Q(A_n)$ also converge to 0? Can you please provide proof or counter example for it. Thanks

Answer 1

Yes, $Q(A)_n$ converge to $0$

Suppose $\frac{dQ}{dP} = f$, with $f>0$ almost surely and $E^P(f) = 1$. Then since $E^P(|f|)< +\infty$, we know for any $\epsilon >0$, $\exists \delta>0$ such that we $E^P(|f|1_{A}) < \epsilon$ for any $P(A)< \delta$.

Then note that $Q(A)= E^P(f1_{A})$, so as long as $P(A)< \delta$ we have $Q(A)< \epsilon$. Then it's easy to see $Q(A_n) \to 0$