Q:

Name for decomposition of joint probability into product of conditional probabilities

Consider a set of events $A_1, \dots, A_n$.
By definition we have
$$\mathbb{P}[A_1 \cap (A_2 \cap \dots \cap A_n)] =  \mathbb{P} [(A_2 \cap \dots \cap A_n) | A_1] \mathbb{P}(A_1).$$
Applying the definition again, 
$$ \mathbb{P} [A_2 \cap (A_3 \cap \dots \cap A_n) | A_1] = \mathbb{P} [ (A_3 \cap \dots \cap A_n) | A_1, A_2] P(A_2|A_1),$$
which together  with the first equality implies
$$\mathbb{P}[A_1 \cap (A_2 \cap \dots \cap A_n)] =  \mathbb{P} [ (A_3 \cap \dots \cap A_n) | A_1, A_2] \mathbb{P}(A_2|A_1) \mathbb{P}(A_1).$$
Continuing the argument recursively yields the following often used decomposition:
$$\mathbb{P}[A_1 \cap A_2 \cap \dots \cap A_n] = \mathbb{P}(A_1) \mathbb{P}(A_2|A_1)  \dots \mathbb{P}(A_{n}|A_1, A_2, \dots, A_{n-1}) .$$
My question is, is there a standard terminology to refer to this decomposition (say as standard as the "Law of Total Probability" terminology)?  

A:

That is repetitive implementation of the definition of conditional probability as you describe, and is referred as Chain Rule.  More generally, it can be written as
$$P\left(\bigcap_{k=1}^n A_k\right)=\prod_{k=1}^n P\left(A_k\bigg\vert\bigcap_{m=1}^{k-1}A_j\right)$$

