Let $\mathcal{E}$ be a random experiment with probability model $(\Omega, \mathcal{F}, P)$. Given $\mathcal{G}=\sigma(G_1,G_2,...) \subset \mathcal F$, a sub $\sigma-$algebra generated by a partition. Fix $A \in \mathcal F$. We know that $P(A| \mathcal G)$ is a random variable such that:
- $P(A| \mathcal G)$ is $\mathcal G-$mensurable
- $\int_G P(A| \mathcal G)(\omega) dP(\omega) = P(A\cap G)$, for all $G \in \mathcal G$.
I know well that $\mathcal G$ can be considered as another experiment and an observer knows if $\omega \in G_i$ for each $i=1,2,3,...$. In this case: $$P(A| \mathcal G)(\omega) = P(A \cap G_i)/P(G_i) \quad \hbox{if } \omega \in G_i$$ Thus, for any $\omega \in \Omega$, I know the interpretation of $P(A| \mathcal G)(\omega)$.
But what is the concrete interpretation of $P(A| \mathcal G)(\omega)$ in the case of $\mathcal G$ is not generated by any partition?