TITLE: When does pairwise independence imply independence?
QUESTION [10 upvotes]: We know if a collection of events are independent, then they are pairwise independent. In general, the converse is not true. However, I'm wondering if there's a condition under which the converse holds. I haven't been able to find anything on this. Any help is appreciated.

REPLY [2 votes]: I was also wondering the same, and I think it's instructive to just restate mutual independence in an equivalent way. I will do it below for 3 real random variables but the statement is generalizable.
Let $X, Y, Z$ be pairwise independent random variables and also assume that $(X,Y)$ is independent of $Z$. Then $X,Y,Z$ are mutually independent.
To prove this, just pick Borel sets $A, B, C$, then it holds that:
$$
\begin{aligned}
\Pr[ X \in A, Y \in B, Z \in C] &= \Pr[ (X,Y) \in A\times B, Z \in C] \\
&= \Pr[(X,Y) \in A\times B] \Pr[Z \in C] \\
&= \Pr[X \in A,  Y \in B] \Pr[Z \in C] \\
&= \Pr[X \in A] \Pr[X \in B] \Pr[Z \in C]
\end{aligned}
$$