# Definition of independence of random variables.

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).

• @ZacharySelk It's the definition I was provided with. Maybe a definition rather than the definition ;). – Dole Oct 25 '18 at 14:31
• @ZacharySelk That's why the question... This is almost directly cut and pasted. There is some added verbiage "for any measurable events $A_1\in \omega$", "the events $f_1\in A_1$...". – Dole Oct 25 '18 at 14:45
• @ZacharySelk Would you say it's correct with the correction I proposed? (Also a typo in my above comment, $\omega$ should have been $\Omega_1$.) – Dole Oct 25 '18 at 15:02
• Yes. $\qquad\qquad$ – user608030 Oct 25 '18 at 15:10