Let $(\Omega,\mathcal F, \mathbb F,\mathbb P)$ be a filtered probability space under the usual conditions and suppose $\mathbb Q\sim\mathbb P$ is an equivalent probability measure. Let $X$ be a $\mathbb P$-semimartingale and $H$ a predictable process $\mathbb P$-integrable with respect to $X$.
By Girsanov-Meyer theorem we know that $X$ is also a $\mathbb Q$ semimartingale. So we can ask:
1) Is $H$ $Q$-integrable with respect to $X$?
2) If so, is the stochastic integral $\int H dX$ the same under any of the two probabilities?