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).

  • $\begingroup$ @ZacharySelk It's the definition I was provided with. Maybe a definition rather than the definition ;). $\endgroup$ – Dole Oct 25 '18 at 14:31
  • $\begingroup$ @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 $f_1\in A_1$...". $\endgroup$ – Dole Oct 25 '18 at 14:45
  • $\begingroup$ @ZacharySelk Would you say it's correct with the correction I proposed? (Also a typo in my above comment, $\omega$ should have been $\Omega_1$.) $\endgroup$ – Dole Oct 25 '18 at 15:02
  • $\begingroup$ Yes. $\qquad\qquad$ $\endgroup$ – user608030 Oct 25 '18 at 15:10

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