Conditional Expectation

Definition.  $\lambda$ is absolutely continuous (AC) with respect to $\mu$, written $\lambda \ll \mu$, if $\mu(A)=0$ implies $\lambda(A) = 0$.

Theorem 2 Radon-Nikodym Theorem  Let $(\Omega,\mathcal{B},P)$ be the probability space.  Suppose $v$ is a positive bounded measure and $v \ll P$.  Then there exists an integrable random variable $X\in \mathcal{B}$, such that
$$v(E) = \int_E XdP, \quad \forall E \in \mathcal{B}$$ $X$ is a.s. unique ($P$) and is written
$$X=\frac{dv}{dP}.$$ We also write $dv=XdP$

Definition of Conditional Expectation

Suppose $X\in L_1(\Omega,\mathcal{B},P)$ and let $\mathcal{G}\subset \mathcal{B}$ be a sub-$\sigma$-field.  Then there exists a random variable $E(X|\mathcal{G})$, called the conditional expectation of $X$ with respect to $\mathcal{G}$, such that

1. $E(X|\mathcal{G})$ is $\mathcal{G}$-measurable and integrable.
2. For all $G\in\mathcal{G}$ we have $$\int_G XdP = \int_G E(X|\mathcal{G})dP$$

Notes.

1. Definition of conditional probability: Given $(\Omega,\mathcal{B},P)$, a probability space, with $\mathcal{G}$ a sub-$\sigma$-field of $\mathcal{B}$, define $$P(A|\mathcal{G})=E(1_A|\mathcal{G}), \quad A\in \mathcal{B}.$$  Thus $P(A|\mathcal{G})$ is a random variable such that 
1. $P(A|\mathcal{G})$ is $\mathcal{G}$-measurable and integrable.
2. $P(A|\mathcal{G})$ satisfies $$\int_G P(A|\mathcal{G})dP = P(A\cap G), \quad \forall G \in \mathcal{G}.$$
2. Conditioning on random variables: Suppose $\{X_t, t\in T \}$ is a family of random variables defined on $(\Omega,\mathcal{B})$ and indexed by some index set $T$.   Define $$\mathcal{G}:=\sigma(X_t,t\in T)$$ to be the \sigma-field generated by the process $\{X_t, t\in T \}$.  Then define $$E(X|X_t, t\in T)= E(X|\mathcal{G}).$$

Note (1) continues the duality of probability and expectation but seems to place expectation in a somewhat more basic position, since conditional probability is defined in terms of conditional expectation.

Note (2) saves us from having to make separate definitions for $E(X|X_1)$, $E(X|X_1,X_2)$, etc.

Countable partitions Let $\{\Lambda_n, n\ge 1 \}$ be a partition of $\Omega$ so thyat $\Lambda_i \cap \Lambda_j = \emptyset, i\neq j$, and $\sum_n \Lambda_n=\Omega$.  Define
$$\mathcal{G}=\sigma(\Lambda_n, n\ge 1)$$ so that
$$\mathcal{G}=\left\{ \sum_{i\in J}\Lambda_i: J\subset\{1,2,\dots \} \right\}.$$ For $X\in L_1(P)$, define
$$E_{\Lambda_n}(X)=\int XP(d\omega|\Lambda_n)=\int_{\Lambda_n}XdP/P\Lambda_n ,$$ if $P(\Lambda_n)>0$ and $E_{\Lambda_n}(X) = 18$ if $P(\Lambda_n)=0$.  We claim

1. $$E(X|\mathcal{G})\overset{a.s.}{=} \sum_{n=1}^\infty E_{\Lambda_n}(X) 1_{\Lambda_n}$$ and for any $A\in \mathcal{B}$
2. $$P(A|\mathcal{G})\overset{a.s.}{=} \sum_{n=1}^\infty P(A|\Lambda_n)1_{\Lambda_n}$$