In Terrence Tao's notes,he made the following definition
Definition 16: A collection $(X_i)_{i \in A}$ of random variables $X_i$ is said to be jointly independent if satisfies one of the following
the distribution $(X_i)_{i \in A}$ is the product of distributions of the $X_i$.
$$ P( \bigwedge_{i \in B} (X_i \in S_i) ) = \prod_{i \in B} P(X_i \in S_i).$$ for all finite sets $B \subseteq A$, and measurable set $S_i$ of $R_i$.
I don't understand definition 1. What is it written out explicitly? No where does he defines the distribution of $(X_i)_{i \in A}$. Suppose we regard $(X_i)_i : \Omega \rightarrow \prod R_i$. I guess definition 1 is :
Definition' $X_i$ are independent iff for all measurable $S \subseteq \prod R_i$, $$ P((X_i)_i \in S ) = \prod P\Big((X_i) \in \pi_i(S)\Big). $$
This results in two questions.
- I don't know how this makes sense as the product on RHS is infinite.
- How would this be equivalent to the second formulation (note $S$ is not a product set here ).