Let $\mathcal{F}$ and $\mathcal{G}$ be two sigma algebras, and suppose $\mathcal{F} = \sigma(L)$, where $L$ is a random variable.
If we want to check the independence between $\mathcal{F}$ and $\mathcal{G}$, we can check the definition (probability of intersection factorizes) or, as it's easier, check the factorization holds only on a $\pi-$system.
I remeber I read somewhere that it's possible to check the independence in other ways (if I'm not wrong, bounded borel functions were involved).
Any suggestion?