Assume $(X,\mathscr{G})$ is a measurable space where $X$ is a finite set and $\mathscr{G}$ is just the power set. Let $\Omega$ be the infinite product of $X$ and $\mathscr{F}$ be the usual product $\sigma$-algebra. Is $\mathscr{F}$ just the power set of $\Omega$? What if $X$ is countable instead of finite.
Thanks.