Conditional expectation w.r.t. discrete measure Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space, $\mathcal E\subseteq\mathcal F$ a sub-$\sigma$-algebra. Assume that $$\mathbb P=\sum_{i=1}^\infty a_i\delta_{\omega_i}$$ where $a_i\in (0,1),$ $\sum_{i=1}^\infty a_i=1$ and $\delta_{\omega_i}$ denotes the point measure at $\omega_i\in\Omega.$ Then, the (unconditional) expectation can be written as $\mathbb E_\mathbb P[X]=\sum_{i=1} a_i X(\omega_i).$ Is there a expression of the conditional expectation $\mathbb E_\mathbb P[X\mid\mathcal E]$ in terms of the $a_i$'s?
 A: Consider the equivalence relation $R$ on $S=\{\omega_i\mid i\in\mathbb N\}$ defined by $\omega_iR\omega_j$ if there exists no $A$ in $\mathcal E$ such that $\omega_i\in A$ and $\omega_j\notin A$. Then $R$ defines a partition $S/R$ of $S$ into equivalence classes. 
For every equivalence class $C$ in $S/R$, choose some $\omega_i$ in $C$. For every $\omega_j$ not in $C$, there exists some $A_j(C)$ in $\mathcal E$ such that $\omega_i$ is in $A_j(C)$ but not $\omega_j$. Let $A(C)$ denote the intersection of the sets $A_C(j)$ over every $j$ such that $\omega_j$ is not in $C$. For every equivalence class $C$ in $S/R$, $A(C)$ is in $\mathcal E$ and, for every $i$, $\omega_i$ is in $A(C)$ if and only if $\omega_i$ is in $C$. Let $B(C)$ denote $A(C)$ minus the union of $A(C')$ on every equivalence class $C'\ne C$. For every equivalence class $C$ in $S/R$, $B(C)$ is in $\mathcal E$, for every $i$, $\omega_i$ is in $B(C)$ if and only if $\omega_i$ is in $C$, and the events $B(C)$ are disjoint. 
Then,
$$
E[X\mid\mathcal E]=\sum\limits_{C}P[B(C)]^{-1}E[X;B(C)]\mathbf 1_{B(C)},
$$ 
where $E[X;B(C)]=\sum\limits_{i\in\mathbb N}a_iX(\omega_i)\mathbf 1_{\omega_i\in C}$ and $P[B(C)]=\sum\limits_{i\in\mathbb N}a_i\mathbf 1_{\omega_i\in C}$ for every equivalence class $C$.
