Change of measure of conditional expectation How can I prove that:
$E_π [ (dQ_X/dπ)  S (T)| F_t ]= E_{Q_X} [S(T) | F_t]E_π [ dQ_X/dπ  | F_t ]$.
Obviously $E_π [(dQ_X/dπ)  S(T) ]= E_{Q_X} [S(T)]$ I know that much, but how to prove when it is conditioned on $F_t$.
 A: I guess the $Q_X$, $S(T)$ and $\mathcal F_t$ come from a particular context, so we can simplify the notation and prove that if $\mu,\nu$ are two probability measures such that $\nu\ll\mu$ and the random variables $X$ and $\frac{\mathrm d\nu}{\mathrm d\mu}$ are integrable, then 
$$\mathbb E_\mu\left(\frac{\mathrm d\nu}{\mathrm d\mu}\cdot X\mid \mathcal F\right)=\mathbb E_{\nu}(X\mid \mathcal F)\cdot\mathbb E_\mu\left(\frac{\mathrm d\nu}{\mathrm d\mu}\cdot\mid\mathcal F\right).$$
To see that, we go back to the definition of conditional expectation. Let $F\in\mathcal F$. Then 
$$\int_F\mathbb E_\mu\left(\frac{\mathrm d\nu}{\mathrm d\mu}\cdot X\mid \mathcal F\right)\mathrm d\mu=\int_F\frac{\mathrm d\nu}{\mathrm d\mu} X\mathrm d\mu=\int_FX\mathrm d\nu.$$
Now, the trick is to use $\mathcal F$-measurability of $\mathbb E_\nu(X\mid\mathcal F)$ to write 
$$\mathbb E_{\nu}(X\mid \mathcal F)\cdot\mathbb E_\mu\left(\frac{\mathrm d\nu}{\mathrm d\mu}\cdot\mid\mathcal F\right)=\mathbb E_\mu\left(\mathbb E_{\nu}(X\mid \mathcal F)\cdot\frac{\mathrm d\nu}{\mathrm d\mu}\cdot\mid\mathcal F\right),$$
then integrate over $F$ with respect to $\mu$. 
A: Alternatively, note that, for ant $A \in \mathcal{F}_t$, 
\begin{align*}
\int_A \frac{dQ_X}{d\pi}S(T) d\pi &=\int_A S(T) dQ_X\\
&=\int_A E_{Q_X}(S(T)\mid \mathcal{F}_t) dQ_X\\
&=\int_A E_{Q_X}(S(T)\mid \mathcal{F}_t) \frac{dQ_X}{d\pi} d\pi\\
&=\int_A E_{\pi}\left(E_{Q_X}(S(T)\mid \mathcal{F}_t) \frac{dQ_X}{d\pi} \mid \mathcal{F}_t \right)d\pi\\
&=\int_A E_{Q_X}(S(T)\mid \mathcal{F}_t) E_{\pi}\left(\frac{dQ_X}{d\pi} \mid \mathcal{F}_t \right)d\pi.
\end{align*}
Therefore, 
\begin{align*}
E_{\pi}\left(\frac{dQ_X}{d\pi}S(T) \mid \mathcal{F}_t\right) &= E_{Q_X}(S(T)\mid \mathcal{F}_t) E_{\pi}\left(\frac{dQ_X}{d\pi} \mid \mathcal{F}_t \right).
\end{align*}
