Independence

Two random variables $x,y$ are said to be independent for any real numbers $x,y$ if: $\mathbb{P}(\{X\le x\}\cap \{Y\le y\})=\mathbb{P}(\{X\le x\})\mathbb{P}(\{Y\le y\}))$

Two random variables $a,b$ are said to be independent for any sets $A,B$ if: $\mathbb{P}(\{A\in a\}\cap \{B\in b\})=\mathbb{P}(\{A\in a\})\mathbb{P}(\{B\in b\}))$