What is the notation (if any) for series probability inclusion? In statistics, what is the notation to use for an event $A$ in $B$ in $C$ in $D$, etc., where the series may continue for a large number of events? The following works for a few events:
$$A\cap B\cap C\cap D\cap\cdots$$
...However, this can become long and arduous for many events that might be considered. Is there any way to succinctly note a lengthy series like this (something like $\sum$ for a sum or $\prod$ for product series)?
The key here is in looking at multiplication laws, such as the following, and how to best note the series of multiplication steps for each $A_n$ in it:
$$p(A_1\cap \cdots \cap A_n)$$
I'd expect something like the following for "a" terms, but am not sure if this is correct (I don't believe the first term would work out here):
$$\prod_{n=1}^a p(A_1\mid A_1\cap A_n)$$

For $A_1\cap\cdots\cap A_n$ you can write $\bigcap_{k=1}^n A_k$.  This is coded in MathJax and LaTeX as \bigcap, and in a "displayed" as opposed to "inline" context, it puts the subscripts and superscripts directly below and above the symbol, just as with $\sum$, thus:
$$
\bigcap_{k=1}^n A_k
$$
If you want the subscripts and superscripts to be formatted that way in an inline context, as $\displaystyle\bigcap_{k=1}^n A_k$, just put \displaystyle before it (that also affects the size).
You can also write
$$
T=\bigcap_{x\in S} A_x
$$
and that means that $a\in T$ if and only if for all $x\in S$, $a\in A_x$.  The set $S$ need not be countably infinite; it can be uncountably infinite.  For example, the intersection of all open intervals $(-\varepsilon,\varepsilon)$ for $\varepsilon>0$ is
$$
\bigcap_{\varepsilon>0} (-\varepsilon,\varepsilon) = \{0\}.
$$
