Given the process $(Z_{n})_{n\ge0}$ where $Z_{n} = \sum_{j=1}^{Z_{n-1}}N_{jn}$. The $N_{jn}$ represent the number of offspring of the j-th individual of the (n − 1)-st generation and are i.i.d. The offspring distribution is $w$. Now $\mathcal F_{n}$ is the sigma algebra generated by $Z_{0},...,Z_{n}$.
Now i want to show $\Bbb E(Z_{n+1}|\mathcal F_{n})=wZ_{n}$.
Here is my approach: The sets $S_{i}:= \{Z_{i}=k_{i}\}$ with $k_{1},..k_{n} \in\Bbb N$ form a partition therefore I can proceed like this
$\Bbb E(Z_{n+1}|\mathcal F_{n})=\sum_{l=1}^{n} \Bbb E(Z_{n+1}|S_{l})*1_{S_{l}}=\sum_{l} \frac{\Bbb E(Z_{n+1}*1_{S_{l}})}{\Bbb P(S_{l})}*1_{S_{l}}$ and now i have no idea how i should deal with this expression. Can somebody help me pls.