I am looking at how to compute the mgf of a sum $S:=\sum_{i=0}^N X_i$ where the $X_i$'s and the $N$ are random variables. I have found a proof where there is $\phi_S(t)=\mathbb{E}[e^{tS}]=\mathbb{E}[\mathbb{E}[e^{tS}|\sigma(N)]]=\sum_m ^\infty \mathbb{P}(N=m)\mathbb{E}[e^{t\sum_i ^m X_i}]$
where $\sigma(N)$ is the sigma algebra generated by $N$. However I can't understand how we can go from the conditional probability to the last sum, can you help me? Thanks
It is genaerally true that $E[X|Y]$ is of the form $f(Y)$ with $f: \mathbb R \to \mathbb R$ measurable. So $Ee^{ts}|\sigma (N)]$ (or $E[e^{ts}|N]$) is of the form $f(N)$ and $Ef(N)=\sum P(N=m) f(m)$.