I am given the following definition of independence of random variables.
Let $f_1: \Omega \rightarrow \Omega_1$, $f_2: \Omega \rightarrow \Omega_2$. The random variables are independent if for any $A_1\in\Omega_1$, $A_2\in\Omega_2$ the events $f_1\in A_1$, $f_2\in A_2$ are independent.
My question, is this correct or should not $A_1\in F_1$ (i.e. the sigma algebra).