I am told that this is a consequence of the Girsanov theorem, yet I do not see how it it is.
Consider some standard model with $dS_i = \mu S_i dt + \sigma S_i dW^P$. Let $Q$ be an equivalent martingale measure. Then, it is claimed that due to the Girsanov theorem, $dS_i = \sigma S_i dW^Q$.
However, the Girsanov theorem only proves this for a particular measure $Q$ which it defines by first introducing a particular variable $L$ defined in terms of another process called the kernel. The $Q$ which is defined through this process may be very different than the $Q$ we have given above, so I don't understand how the Girsanov theorem can be used?
Should we not instead prove that given any martingale measure $Q$, then we can always determine a kernel which can be used to define this measure $Q$ using the recipe in the Girsanov theorem, and THEN we can use Girsanov?