Joint probability distribution only measures product sets?

According to these notes (top of p 133), "We say that random variables $X_1, X_2, \ldots X_n : \Omega \to \mathbb{R}$ are jointly continuous if there is a joint probability density function $p(x_1, x_2, \ldots, x_n)$ such that $$ P(\{X_1 \in A_1, X_2 \in A_2,\ldots, X_n \in A_n\}) = \int_A p (x_1, x_2,\ldots, x_n) dx_1dx_2\ldots dx_n. $$ where $A = A_1 \times A_2 \times \cdots \times A_n$."

This seems like we can only measure "rectangle" Borel subsets of $\mathbb{R}^n$. What about sets like $\{X < Y\}$? It seems like I wouldn't be able to measure such sets...Clearly, I'm misunderstanding something.

Answer

Let's try to define the solution first $$ P[\{X< Y\}] = \int_{-\infty}^\infty \int_{-\infty}^y f_{(X,Y)}(x,y) dx dy, $$ and the set over which we integrate can not be written as the product of two sets as defined there. So: we can measure other sets as well. We can calculate probabilities but the author seems to use a definition of continuous random variables that if $A$ is product of sets then there is a density (p).

I think this statement is just the definition of the density of continuous rvs and not a statement about which sets we can measure.

The way to read it is if $A$ is of the product form and if we can write the above formula with some $p$, then we call $p$ a density. That's the statement.