$\Theta \rho ϵ\eta \Pi \alpha \tau \pi$

Suppose that $0\le X_1\le X_2\le \cdots$ are random variables and $X\left(\omega \right)={\lim }_{n\to \infty }{X}_n\left(\omega \right)$. Then $$E\left(X\right)=\underset{n\to \infty }{\lim }E\left(X_n\right).$$

Suppose that $X_1,X_2,\dots$ are random variables that converge pointwise to $X$, and suppose that $\left|X_n\right|\le Y$ for every $n$, where $E\left(Y\right)<\infty$. Then $$E\left(X\right)=\underset{n\to \infty }{\lim }E\left(X_n\right).$$

Suppose that $X_1,X_2,\dots$ are independent identically distributed random variables with finite expected value $\mu$. Then $$\frac{1}{n}\sum _{k=1}^n{X}_k\to \mu$$ in probability.

Suppose that $X_1,X_2,\dots$ are independent identically distributed random variables with finite expected value $\mu$. Then $$\frac{1}{n}\sum _{k=1}^n{X}_k\to \mu$$ almost surely.