Disjoint groups of an independent sigma-algebra family remain independent

For each $r<m$, let ${\mathrm{\Pi }}_r$ be the class consisting of $\mathrm{\Omega }$ together with all finite intersections ${\bigcap }_{i\in F}{A}_i$, where $F\subseteq J_r$ is finite and $A_i\in {\mathcal{F}}_i$ for every $i\in F$. Each ${\mathrm{\Pi }}_r$ contains $\mathrm{\Omega }$ and is closed under finite intersections, so it is a pi-system. Moreover $\sigma ({\mathrm{\Pi }}_r)={\mathcal{G}}_r$ by definition of ${\mathcal{G}}_r$.

Fix $C_r\in {\mathrm{\Pi }}_r$ for each $r<m$. Because the index sets $J_r$ are pairwise disjoint, the event ${\bigcap }_{r<m}{C}_r$ is a finite intersection of events taken from distinct members of the original independent family. Hence [L1] gives $\mathbb{P}\left({\bigcap }_{r<m}{C}_r\right)={\prod }_{r<m}\mathbb{P}(C_r)\mathrm{.}$

Step 1.2 says that the pi-systems ${\mathrm{\Pi }}_0,\dots ,{\mathrm{\Pi }}_{m-1}$ are independent. Applying [L2] and using step 1.1 yields independence of $\sigma ({\mathrm{\Pi }}_r)={\mathcal{G}}_r$ for every $r<m$.