What is $\mathbb P\{W\leq x,Y\leq y\}$ ?Is it $\mathbb P\{\omega \in \Omega : X(\omega )\leq x,Y(\omega )\leq y\}$?

Question

Let $(\Omega ,\mathbb F,\mathbb P)$. I'm confuse about something : Let $X,Y:\Omega \to \mathbb R$ r.v. What is $$\mathbb P\{X\leq x,Y\leq y\} \ \ ?\tag{1}$$

Is it, $$\mathbb P\{\omega \in \Omega \mid X(\omega )\leq x,Y(\omega )\leq y\}$$ or $$\mathbb P\{(\omega ,\omega ')\in \Omega ^2\mid X(\omega )\leq x,Y(\omega ')\leq y\} \ \ ?\tag{2}$$

I say the notation $$\mathbb P\{X\leq x,Y\leq y\}=\mathbb P\{\{\omega \in \Omega \mid X(\omega )\leq x\}\cap \{\omega \in \Omega \mid Y(\omega )\leq x\}\},$$

but in otherhand, I know that $\mathbb P\{X\in A,Y\in B\}$ is a product measure on $\mathbb R^2$, so may be we also have a product measure on $\Omega ^2$ ? I'm a bit confuse... Espacially that if $(\Omega ',\mathcal F',\mathbb P')$, and $X:\Omega \to \mathbb R$, $Y:\Omega '\to \mathbb R$, then $$\mathbb P\otimes \mathbb P'\{X\leq x, Y\leq y\}=\mathbb P\otimes\mathbb P'\{(\omega ,\omega ')\in \Omega \times \Omega '\mid X(\omega )\leq x, Y(\omega ' )\leq y\}=\mathbb P\{X\leq x\}\mathbb P'\{Y\leq y\},$$ which is a measure on $\mathbb R^2$. That's why I would think that $(é)$ is true but not $(1)$. What do you think ?

Answer 1

The correct interpretation is the first one: $\mathbb P\{X\le x,Y\le y\}=\mathbb P\{\omega \in \Omega \mid X(\omega )\leq x,Y(\omega )\leq y\}$. You say

I know that $P(X\in A,Y\in B)$ is a product measure on $\mathbb R^2$,

but this is untrue in general. This is only true when $X$ and $Y$ are independent.

The interpretation $\mathbb P\{(\omega ,\omega ')\in \Omega ^2\mid X(\omega )\leq x,Y(\omega ')\leq y\}$ cannot be correct, since $\mathbb P(A)$ is defined for measurable subsets of $\Omega$, not subsets of $\Omega^2$.

Answer 2

The product measure is in no way involved here. Product measure makes its appearance only when $X$ and $Y$ are independent. The correct expression is $P\{\omega \in \Omega: X(\omega) \leq x,Y(\omega) \leq y\}$.