Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. If $X$ is a random variable, what is meant by $\mathbb{P}(X\le{t})$ or $\mathbb{P}(X\ge{t})$ for some t?
The context I am confused about is in a problem using chernoff bounds. why can we do:
$$\mathbb{P}(|S_N-\mu|\ge{\delta{}\mu})=\mathbb{P}(S_N-\mu\ge{\delta{}\mu})+\mathbb{P}(S_N-\mu\le{-\delta{}\mu})$$
and what is really meant by the probability measure of these inequalities?
edit: I'm essentially looking for an explicit definition of $\mathbb{P}(X\le{t})$.
$\mathbb{P}(X\le{t})$ is a shorthand for $\mathbb{P}(\{\omega \in\Omega\mid X (\omega) \le{t}\})$. Thus, $$\mathbb{P}(|S_N-\mu|\ge{\delta{}\mu})=\mathbb{P}([S_N-\mu\ge{\delta{}\mu}]\cup [S_N-\mu\le{-\delta{}\mu}])=\mathbb{P}(S_N-\mu\ge{\delta{}\mu})+ \mathbb{P}(S_N-\mu\le{-\delta{}\mu}).$$