I am currently working through the results presented in the paper titled "Consistent Price Systems and Face-lifting Pricing under Transaction Costs" and authored by P. Guasoni, M. Rásonyi and W. Schachermayer. (I believe this link should be accessible: Link)
Let $(\Omega, \mathscr{g}, \text{{$\mathscr{g_n}$}$_{n \in \mathbb{N}}$}, \mathbb{P})$ be a filtered probability space with: $\mathscr{g}_0$ the trivial sigma algebra and $\mathscr{g}$ the sigma algebra generated by all $\text{{$\mathscr{g_n}$}}$.
Let {$Z_n$} be an adapted process ($Z_n$ is $\mathscr{g}_n$-measurable), {$\alpha_n$} a predictable process ($\alpha_n$ is $\mathscr{g}_{n-1}$-measurable). Lastly we have a collection of increasing sets $\text{{${A_n}$}$_{n \in \mathbb{N}}$}$ increasing to $\Omega$ $\mathbb{P}$ a.s with $A_0 = \emptyset$ and $A_n \in \mathscr{g}_{n}$.
We have that these processes / sets satisfy the following identity:
\begin{align} \mathbb{E}[Z_{n} 1_{\Omega \setminus A_n}|\mathscr{g}_{n-1}] = 1_{\Omega \setminus A_{n-1}}(1-\alpha_n) \tag 1 \end{align}
The main question I had is about the following application of iterated conditional expectations to obtain:
\begin{align} \mathbb{E}[\prod_{i=1}^n(Z_i 1_{\Omega \setminus A_i})]= \\ \mathbb{E}[(1-\alpha_n)1_{\Omega \setminus A_{n-1}}\prod_{i=1}^{n-1}(Z_i 1_{\Omega \setminus A_i})]= &&\text{conditioning on $\mathscr{g}_{n-1}$ and applying (1)} \\ ...= &&\text{(this is the step I struggle with)} \\ \mathbb{E}[\prod_{i=1}^n(1-\alpha_i)] \tag 2 \end{align}
My first thought was to simply condition on $\mathscr{g}_{n-2}$ to further simplify the expectation but I run into the following issue:
\begin{align} \mathbb{E}[\mathbb{E}[(1-\alpha_n)1_{\Omega \setminus A_{n-1}}\prod_{i=1}^{n-1}(Z_i 1_{\Omega \setminus A_i})|\mathscr{g}_{n-2}]]= &&\text{conditioning on $\mathscr{g}_{n-2}$} \\ \mathbb{E}[\prod_{i=1}^{n-2}(Z_i 1_{\Omega \setminus A_i})\mathbb{E}[(1-\alpha_n)1_{\Omega \setminus A_{n-1}}Z_{n-1}|\mathscr{g}_{n-2}]] &&\text{$Z_i 1_{\Omega \setminus A_i}$ being $\mathscr{g}_{n-2}$-measurable for $i \le n-2$} \\ \end{align}
What I would like to do is apply the identity (1) to the remaining conditional expectation but I am unsure how to take the $(1-\alpha_n)$ factor outside of the conditional expectation as it is $\mathscr{g}_{n-1}$-measurable.
I am looking for any advice or hints on how get to the final expectation (2). I can provide further details on the context but I believe the snips below for the definition used and the proof until (2) should be everything.
