What Pi-System is used to define independence for random variables

Question

Independent random variables [ edit]
The theory of $\pi$-system plays an important role in the probabilistic notion of independence. If $X$ and $Y$ are two random variables defined on the same probability space $(\Omega, \mathcal{F}, \mathrm{P})$ then the random variables are independent if and only if their $\pi$-systems $\mathcal{I}_{X}, \mathcal{I}_{Y}$ satisfy
$\mathrm{P}[A \cap B]=\mathrm{P}[A] \mathrm{P}[B] \quad$ for all $A \in \mathcal{I}_{X}$ and $B \in \mathcal{I}_{Y}$
which is to say that $\mathcal{I}_{X}, \mathcal{I}_{Y}$ are independent. This actually is a special case of the use of $\pi$-systems for determining the distribution of $(X, Y)$.

There are many Pi-Systems for a set $\Omega.$ Which one do we mean when we define independence of random variables?

Answer 1

The quote appears to be from Wikipedia: https://en.wikipedia.org/wiki/Pi-system#Independent_random_variables

Looking in earlier sections, we see $\mathcal{I}_f$ is defined under Examples

For any measurable function $f: \Omega \to \mathbb{R}$, the set $\mathcal{I}_f = \{f^{-1}((-\infty,x]) : x \in \mathbb{R}\}$ defines a $\pi$-system, and is called the $\pi$-system generated by $f$.

So in words, given a real-valued random variable $X$ on probability space $(\Omega, \mathcal{F}, P)$, each subset of $\Omega$ which is in the $\pi$-system $\mathcal{I}_X$ is the set of all outcomes which give a value of $X$ not larger than a specific upper bound.