Suppose we have random variables $X:(\Omega, \mathcal{F}, \mathbb{P}) \rightarrow (\Omega_{X}, \mathcal{F}_{X})$ and $W:(\Omega, \mathcal{F}, \mathbb{P}) \rightarrow (\mathbb{N}_{n}, \mathcal{P}(\mathbb{N}_n))$ where $n \in \mathbb{N}$ and $\mathbb{N}_n = \mathbb{N} \cap [1, n]$.
I can see a way of defining $X \vert W=w$, which I will denote by $X_w$, for $w \in \mathbb{N}_n$.
Let $E_w = W^{-1}(\{w\})$ and $\mathcal{F}_w = \{A \cap E_{w}:A \in \mathcal{F}\}$. Assume that $\mathbb{P}(E_w) > 0$. Define $\mathbb{P_w: \mathcal{F}_{w} \rightarrow [0, 1]}, A \mapsto \mathbb{P}(A \vert E_w)$.
One defines $X_w:(E_w, \mathcal{F}_w, \mathbb{P}_w) \rightarrow (\Omega_X, \mathcal{F}_X), x \mapsto X(x)$.
This seems to be a "natural" way to define this conditional random variable, but its causing me some unexpected surprises in my work. It may be the case that the surprises are just something I will have to deal with.
Has anyone seen an alternative way to think about this?