I am trying to understand the notation $P(X\in C, Y\in D)$ If $X$ and $Y$ are defined on the same probability space then is this equivalent to $P(\{\omega\in \Omega:X(\omega)\in C, Y(\omega)\in D\})$
or is it the probability measure on a product space.
$$P(\{(\omega_1, \omega_2)\in \Omega\times \Omega:X(\omega_1)\in C, Y(\omega_2)\in D\})$$
where P is a product measure. Which makes no sense to me since they should explicitly define it as a product measure or is it implied by the joint random. Basically, does this notation
$P(X\in C, Y\in D)$ always imply the intersection definition $P(\{\omega\in \Omega:X(\omega)\in C, Y(\omega)\in D\})$. If X and Y are defined on different spaces then $P(X\in C, Y\in D)$ makes no sense and a product measure should be defined?
Best Answer
Your first interpretation is correct. If $X$ and $Y$ appear in the same event then they must be defined on the same probability space. The convention is: $$ \{X\in C, Y\in D\} := \{ \omega\in \Omega: X(\omega)\in C , Y(\omega)\in D\} $$ and the comma appearing in the RHS is shorthand for 'and'. Thus we also have: $$\{X\in C, Y\in D\} =\{\omega\in\Omega:X(\omega)\in C\}\cap\{\omega\in\Omega:Y(\omega)\in D\} $$