Consider the case $\Omega = \mathbb R^6 , F= B(\mathbb R^6)$ Then the projections $\ X_i(\omega) = x_i ,[ \omega=(x_1,x_2,\ldots,x_6) \in \Omega $ are random variables $i=1,\ldots,6$. Fix $\ S_n = S_0$ $\ u^{\Sigma X_i(\omega)}d^{n-\Sigma X_i(\omega)} \omega \in \Omega $, $\ n=1,\ldots,6 $.
Choose the measure P = $\bigotimes_{i=1}^6 Q$ on ($\Omega,F$) where $Q$ denotes the measure $p\delta_1 + q\delta_0 $ on $(\mathbb R, B(\mathbb R))$ for some $p,q>0$ such that $p+q = 1$. Show that the projections $\ X_i(\omega), i=1,\ldots,6$ are mutually independent.
Since $\ X_i(\omega)$ is a random variable then am I correct in saying that to show their independence I must show that their sigma algebras $\sigma(\ X_i(\omega))$ are independent how would I go about doing this?
Thanks very much!