Conditional expectation

Conditional expectation given a sigma-algebra \(\mathcal{G}\)

\[\forall G \in \mathcal{G}, \quad \int_G X(w) dP(w) = \int_G E[X|\mathcal{G}](w)dP(w)\]

Conditional expectation expressed by simple functions

For simplicity and without loss of generality, I limit the situation to be \(\mathcal{G}=(\emptyset, G, G^\complement, \Omega)\).

\[E[X|\mathcal{G}](\omega)=E[X|G]1_{G}(\omega)+E[X|G^\complement]1_{G^\complement}(\omega)\tag{1}\]

Note that \(E[X|\mathcal{G}](w)\) is a function of \(\omega \in \Omega\) and \(E[X|G], E[X|G^\complement]\) are constants.

Conditional expectation given a set \(G\)

Set a sigma-algebra generated by the set \(G\) as \(\mathcal{G}=(\emptyset, G, G^\complement, \Omega)\) then

\[E[X|G]=\frac{E[X1_{G}]}{P(G)}\]

holds by intergrating both sides of Equation \((1)\) with respect to the set \(G\).