joint probability distribution

given two random variables that are defined on the same probability space, the joint probability distribution is the corresponding probability distribution on all possible pairs of outputs. it is denoted by $\text{[math]}$

$\text{[math]}$

klenke's definition is rooted in measure theory:

for any $\text{[math]}$ , let $\text{[math]}$ be a real random variable. for any finite subset $\text{[math]}$ , let
$\text{[math]}$ then $\text{[math]}$ is called the joint distribution function of $\text{[math]}$ . the probability measure $\text{[math]}$ on $\text{[math]}$ is called the joint distribution of $\text{[math]}$ .
[cite:@klenke_prob_2020 definition 2.20]