Random variable

Definition (review)
A random variable is a measurable function (where $C$ is given with the Borel sigma-algebra generated by the Euclidean topology):

f : Ω \mapsto C

such that

f^{- 1} (A) \in Σ

for every open set $A = {z \in C : a < R e [z] < b}$ and

s u p_{Ω} {f (ω)} < \infty .

$◼$

A random variable which only takes the values 0 or 1 encodes the same data as an event (more precisely, it is the indicator function $χ_{E}$ of a unique event, which takes the value $1$ on $E$ and $0$ on its complement). More generally, we can construct events from random variables: for any Borel subset $E ubseteq athbb{R}$ the preimage $X^{-1}(E)$ is an event (the event that $X$ lies in $E$ ), often written $X n E$ , and so we can consider its probability $athbb{P}(X n E)$ .