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. \[P(A|\mathcal{G})\overset{a.s.}{=} \sum_{n=1}^\infty P(A|\Lambda_n)1_{\Lambda_n}\]