What does $\mathbb{P}[X=a,Y=b]$ mean, notationally?

Question

I am wondering what the notation $\mathbb{P}[X=a,Y=b]$ means- for random variables $X$ and $Y$, where $a$ and $b$ are values they respectively take on- I know that $\mathbb{P}[X=a]$ is the probability of the event equivalent to the set of all sample points for which $X=a$, but am unsure what it means for there to be multiple arguments to the probability function in this manner.

Answer 1

There's an implied conjunction (or intersection of events). In other words, it means the probability that both $X=a$ and $Y=a.$ Being that $P(X=a)$ is the measure of $X^{-1}(\{a\})$ and $P(Y=b)$ is the measure of $Y^{-1}(\{b\}),$ $P(X=a,Y=b)$ is the measure of $X^{-1}(\{a\})\cap Y^{-1}(\{b\}).$