Convergence Theorems

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Let $X$ and $\{X_n\}$ be random variables.

1. Monotone Convergence Theorem
If $X_n\geq0$ and $X_n\uparrow X$ a.e. on $\Lambda$, then $$\underset{n\rightarrow\infty}{\lim}\int_\Lambda X_n\,d\mathscr{P}=\int_\Lambda X\,d\mathscr{P}=\int_\Lambda\underset{n\rightarrow\infty}{\lim}X_n\,d\mathscr{P}.$$


2. Dominated Convergence Theorem
If $\underset{n\rightarrow\infty}{\lim}X_n=X$ a.e. and, for all $n$, $|X_n|\leq Y$ a.e. on $\Lambda$ with $\mathscr{E}(Y)<\infty$, then $$\underset{n\rightarrow\infty}{\lim}\int_\Lambda X_n\,d\mathscr{P}=\int_\Lambda X\,d\mathscr{P}=\int_\Lambda\underset{n\rightarrow\infty}{\lim}X_n\,d\mathscr{P}.$$


3. Bounded Convergence Theorem
If $\underset{n\rightarrow\infty}{\lim}X_n=X$ a.e. and there exists a constant $M$ such that, for all $n$, $|X_n|\leq M$ a.e. on $\Lambda$, then $$\underset{n\rightarrow\infty}{\lim}\int_\Lambda X_n\,d\mathscr{P}=\int_\Lambda X\,d\mathscr{P}=\int_\Lambda\underset{n\rightarrow\infty}{\lim}X_n\,d\mathscr{P}.$$


4. Fatou's Lemma
If $|X_n|\geq0$ a.e. on $\Lambda$, then $$\int_\Lambda\underset{n\rightarrow\infty}{\liminf}X_n\,d\mathscr{P}\leq\underset{n\rightarrow\infty}{\liminf}\int_\Lambda X_n\,d\mathscr{P}.$$Furthermore, if for all $n$, $|X_n|\leq Y$ a.e. on $\Lambda$ with $\mathscr{E}(Y)<\infty$, the above remains true as well as $$\int_\Lambda\underset{n\rightarrow\infty}{\limsup}X_n\,d\mathscr{P}\geq\underset{n\rightarrow\infty}{\limsup}\int_\Lambda X_n\,d\mathscr{P}.$$See Proof of Fatou's Lemma
See Application of Fatou's Lemma