Law of total expectation and Information. I am reading a paper by Glostom and Milgrom on "Bid/Ask transaction prices". Here is the gist of the issue I am having.  $F_t$ is the current information / filtration. $Z_t$ is a random variable. $A$ is the ask price. $V$ is the random unknown fair value of a stock.  The claim is below.

$E[V 1_{\{Z_t &gt; A\}} | S_t] = E[V | S_t, \{Z_t &gt; A\}] P(Z_t &gt; A | S_t)$

I kinda see that this is the law of total expectation. I have seen and know how to prove when the conditioning is over single sets, I am not sure how I can extend a measure theoretic argument over to this situation. In general how does one prove this to conditioning over any set of information sets.
What is $S_t$ in this situation?

Edit: If $S_t$ is also a filtration and $Z_t$ is $S_t$-measurable I don't really see what the conditioning $S_t,\{Z_t&gt;A\}$ is supposed to mean. I guess they probably mean the random variable $Y=E[V|S_t,\{Z_t&gt;A\}]$ which is $0$ outside of $\{Z_t&gt;A\}$ and equal to $c\cdot E[V|S_t]$ inside of $\{Z_t&gt;A\}$. Then the question of how to choose this $c$ value is directly related to your question. $c$ needs to make $Y$ a random variable, i.e. $E[Y]=1$, and for that you need to inflate it to account for the "lost" expectation by setting $Y=0$ outside of $\{Z_t&gt;A\}$. Therefore $c=\frac{1}{P(Z_t&gt;A|S_t)}$. If you enter this into your equation you will see that the equation holds.
What is $S_t$ in this situation?

Edit: If $S_t$ is also a filtration and $Z_t$ is $S_t$-measurable I don't really see what the conditioning $S_t,\{Z_t&gt;A\}$ is supposed to mean. I guess they probably mean the random variable $Y=E[V|S_t,\{Z_t&gt;A\}]$ which is $0$ outside of $\{Z_t&gt;A\}$ and equal to $c\cdot E[V|S_t]$ inside of $\{Z_t&gt;A\}$. Then the question of how to choose this $c$ value is directly related to your question. $c$ needs to make $Y$ a random variable, i.e. $E[Y]=1$, and for that you need to inflate it to account for the "lost" expectation by setting $Y=0$ outside of $\{Z_t&gt;A\}$. Therefore $c=\frac{1}{P(Z_t&gt;A|S_t)}$. If you enter this into your equation you will see that the equation holds.