Independence of random variables

A family of random variables $(X_i)_{i\in I}$ on a probability space $(\Omega,\mathcal F,\mathbb P)$ is independent if for every finite choice of indices $i_1,\dots,i_k\in I$ and every choice of Borel sets $A_1,\dots,A_k\subseteq\mathbb R$,

This says that all events of the form $\{X_i\in A\}$ behave like independent events under probability . Equivalently, the sigma-algebras $\sigma(X_i)$ generated by the variables are independent .

Examples:

• Let $\Omega=\{0,1\}^2$ with $\mathbb P$ uniform, and define $X(\omega_1,\omega_2)=\omega_1$, $Y(\omega_1,\omega_2)=\omega_2$. Then $X$ and $Y$ are independent random variables .
• Let $\Omega=[0,1]^2$ with the product Lebesgue measure (normalized to a probability measure), and set $X(u,v)=u$, $Y(u,v)=v$. Then $X$ and $Y$ are independent and each has the uniform distribution on $[0,1]$.