Proposition (Properties of Expectation)

Let $(\Omega,\mathcal{F},P)$ be a Probability Space.

1. Linearity: $\forall X,Y\in\mathscr{L}^{1}(\Omega,\mathcal{F},P)$$$E[\alpha X+\beta y]=\alpha E[X]+\beta E[Y]$$
2. Monotonicity: $\forall X\in\mathscr{L}^{1}(\Omega,\mathcal{F},P)$ $$X\ge0\text{ a.s.}\implies E[X]\ge0\text{ a.s.}$$
3. MCT: Let $(X_{n})_{n\in\mathbb{N}}$ be a sequence of rvs where $X_{n}\ge0,\forall n\in\mathbb{N}$ and $X_{n}\uparrow X$ as $n\to\infty$ then $$E[X_{n}]\uparrow E[X]\text{ as }n\to\infty$$
4. Fatou’s Lemma: Let $(X_{n})_{n\in\mathbb{N}}$ be rv where $X_{n}\ge0,\forall n\in\mathbb{N}$ then $$E\left[\liminf_{n \to \infty }X_{n} \right]\le \liminf_{ n \to \infty } E[X_{n}]$$
5. Dominated Convergence Theorem: Let $(X_{n})_{n\in\mathbb{N}}\subset \mathscr{L}^{1}(\Omega,\mathcal{F},P)$, let $Y\in\mathscr{L}^{1}(\Omega,\mathcal{F},P)$ and assume $|X_{n}|\le Y, \ \forall n\in\mathbb{N}$. Let $X:\Omega\to \mathbb{R}$ s.t. $X_{n}\to X$. Let $\mathcal{G}\subset \mathcal{F}$ sub-σ-algebra. Then $X\in \mathscr{L}^{1}$and $$E[X_{n}|\mathcal{G}]\xrightarrow{L^{1}} E[X|\mathcal{G}]$$