Assume $(\Omega,\mathcal{F}, \mathbb{P})$ is a probability space, $\{F_t\}_{t\leq T}$ is an adapted filteration and $M_t$ is a martingale with respect to that with $M_0=1$. We can define another probability measure on $F_T$ using the following radon nikodym derivative:

\begin{align} \frac{d\mathbb{Q}}{d\mathbb{P}}=M_T \end{align}

Assume $T_0<T$ and $A\in F_{T_0}$ is a measure zero event (w.r.t $\mathbb{P}$). I have problem finding the Radon Nikodym derivative of $\mathbb{Q}(\cdot|A)$ with respect to $\mathbb{P}(\cdot|A)$ in terms of the martingale $M_t$.

My guess is the answer should be $\frac{M_T}{M_{T_0}}$, but I have problem working with terms like $\mathbb{Q}(A)/\mathbb{P}(A)$, etc.

I understand that the problem might not be well defined for all $A$. So I also would like to know what conditions are required for $A$? and given those conditions are satisfied, what is the answer to this problem?


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