Ive found a statement of Girsarnovs theorem that looks as follows
"Every $P$-semimartingale is a $Q$ semimartingale, in particular if $M$ is a local martingale then $\hat{M}_{t}=M_{t}-D_{t}^{-1}[M,D]_{t}$ is a $Q$ local martingale. Let $A_{t}=D_{t}^{-1}[M,D]_{t}$
Lets say $X=M_{1}+B$ is the semimartingale decomposition w.r.t $P$, do we just add and remove the above $A$, to get $X=M_{1}-A+A+B$ to get a $Q$ semimartingale?
How is X written as a $Q$ semimartingale? Or how do we see that it is a $Q$ semimartingale?
Since we also have that $[M,X]=[\hat{M},X]$ are the same for all $X$ both under $\mathbb{P}$ and $\mathbb{Q}$ we know that $M$ and $\hat{M}$ are the same up to a constant. Therefore $X=\hat{M}+B$ is the $\mathbb{Q}$-semimartingale form