Probability Space, Event, and Random Variable

A probability space is a measure space $(\Omega,\mathcal{F},P)$ whose measure $P$ is a probability measure in the sense of that definition, that is, $P(\Omega)=1$. Members of $\mathcal{F}$ are called events, and $P(A)$ is the probability of the event $A$.

A random variable on $(\Omega,\mathcal{F},P)$ is a measurable function $X:\Omega\to\mathbb{R}$ (with respect to $\mathcal{F}$ and the Borel $\sigma$-algebra).

For a Borel set $B$ one writes $\{X\in B\}$ for the event $X^{-1}(B)$, and analogously $\{X\le t\}$, $\{X>t\}$, $\{|X-c|\ge a\}$, and so on, for the preimages of the corresponding Borel sets; probabilities of such events are written $P(X\in B)$, $P(X\le t)$, etc.