Theorem

basic criteria for independence

If for each $i = 1, \dots , n$ we have $\mathcal{C}_i$ is a non-empty class of events such that

1. $\mathcal{C}_i$ is a $\pi$-system
2. $\mathcal{C}_i$, $i = 1, \dots , n$ are independent

then $\sigma(\mathcal{C}_1), \dots , \sigma(\mathcal{C}_n)$ are independent.