Suppose we have a probability space $(\Omega, \mathcal{F}, P)$ and $\mathcal{F}$-measurable random variables $X$ and $Z$ on this space. Suppose $Z$ is discrete and let $\mathcal{G}=\sigma(Z)$

The random variable $\mathbf{E}[X \mid Z]$ is the same as the random variable $\mathbf{E}[X \mid \mathcal{G}]$. But now suppose $A$ is an event, then $\mathbf{E}[X \mid A, Z]$ is interpreted to be a random variable, which is a function $f(Z)$ of $Z$. When $Z=z$, it has value $\mathbf{E}[X \mid A \cap \{Z=z\}]$.

What would be a good to write this using the sigma algebra notation?

For example, we have $$ \mathbf{E}[X \mid A, Z] =\frac{\mathbf{E}[X1_{A}\mid \mathcal{G}]}{\mathbf{E}[1_A \mid \mathcal{G}] } $$

where, for $\omega \in \Omega$, $1_A(\omega)=1$ if $\omega \in A$ and $1_A(\omega)=0$ otherwise.

The RHS seems of the above is not intuitive. If it was written by itself, it is not clear what it would mean, whereas the LHS is clear.

A first attempt may write $\mathbf{E}[X \mid A \cap \mathcal{G}]$, but $ A \cap \mathcal{G}$ is not a sigma algebra over $\Omega$.

So is there a standard, if not intuitive way to write the random variable $\mathbf{E}[X \mid A, Z]$ using the sigma algebra generated by $Z$?


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