Proof of the Dominated Convergence Theorem
Theorem
Given a probability space $( \Omega , \mathcal{F} , P)$.
If a sequence of random variables $\left\{ X_{n} \right\}_{n \in \mathbb{N}}$ satisfies $n \in \mathbb{N}$ for all $Y \in \mathcal{L}^{1} (\Omega)$ and some $Y \in \mathcal{L}^{1} (\Omega)$, then $$ X_{n} \to X \text{ a.s.} \implies E( X_{n} | \mathcal{G} ) \to \mathcal{G} ) \text{ a.s.} $$
- $\text{a.s.}$ means almost surely.
Description
The Dominated Convergence Theorem for conditional expectations, also known as the Conditional Dominated Convergence Theorem (CDCT), tells us that the Dominated Convergence Theorem (DCT) applies to conditional expectations just as it does in other contexts, playing a similar role in probability theory.
Proof
Properties of conditional expectation
- [7]: $E(X+Y | \mathcal{G}) = E(X | \mathcal{G}) + E(Y| \mathcal{G}) \text{ a.s.}$
- [10]: $\left| E( X | \mathcal{G} ) \right| \le E ( | X | | \mathcal{G} ) \text{ a.s.}$
$$ \begin{align*} & \left| E( X_{n} | \mathcal{G} ) - E( X | \mathcal{G} ) \right| \\ \color{red}{=}& \left| E( X_{n} - X | \mathcal{G} ) \right| \\ \color{blue}{\le}& E( \left| X_{n} - X \right| | \mathcal{G} ) \\ \le & E \left( \sup_{k \ge n} \left| X_{k} - X \right| | \mathcal{G} \right) \end{align*} $$ So, by the Monotone Convergence Theorem for conditional expectations, properties of lim sup, and condition $X_{n} \to X \text{ a.s.}$, $$ \begin{align*} & \lim_{n \to \infty} \left| E( X_{n} | \mathcal{G} ) - E( X | \mathcal{G} ) \right| \\ \le & \lim_{n \to \infty} E( \sup_{k \ge n} \left| X_{k} - X \right| | \mathcal{G} ) \\ \color{red}{=}& E \left( \lim_{n \to \infty} \sup_{k \ge n} \left| X_{k} - X \right| | \mathcal{G} \right) \\ =& E \left( \lim_{n \to \infty} \left| X_{n} - X \right| \mathcal{G} \right) \\ \color{blue}{=}& 0 \text{ a.s.} \end{align*} $$
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