If $(\mathcal F_1,\mathcal F_2,\mathcal F_3)$ is independent, is $\mathcal F_1\vee\mathcal F_2$ independent of $\mathcal F_3$?
Solution 1:
Sets of the form $A \cap B$ with $A \in \mathcal F_1, B \in \mathcal F_2$ form a $\pi$ system which generates $\mathcal F_1 \vee \mathcal F_2$. The equation $P(C\cap D) =P(C)P(D)$ holds if $C$ is above type and $D \in \mathcal F_3$. Apply Dymkin's $\pi -\lambda$ Theorem (with $D \in \mathcal F_3$ fixed) to finish.