Let $X_1, X_2, X_3, \ldots \in \mathscr{L}^1(\mathbb{P})$ be independent, identically distributed random variables with values in $\left[0, \infty\left[\right. \right.$ and $N \in \mathscr{L}^1(\mathbb{P})$ a random variable independent of $X_1, X_2, X_3, \ldots$ with values in $\mathbb{N}$. Let the random variable $S_N$ be given by $$ S_N:=\sum_{n=1}^{\infty} S_n \mathbf{1}_{\{N=n\}} $$ where $S_n:=\sum_{k=1}^n X_k, n \in \mathbb{N}$. Show that $$ \mathbb{E}\left(S_N\right)=\mathbb{E}(N) \mathbb{E}\left(X_1\right) $$ holds.
First of all, I think I have to show that $S_N$ is a random variable. Nevertheless, I got this: $$ \mathbb{E}\left(S_N\right)=\mathbb{E}\left(\sum_{n=1}^{\infty} S_n \mathbf{1}_{\{N=n\}}\right)=\sum_{n=1}^{\infty}\mathbb{E}\left(S_n \mathbf{1}_{\{N=n\}}\right)=\sum_{n=1}^{\infty}\mathbb{E}(S_n) \mathbb{E}(\mathbf{1}_{\{N=n\}}) $$ and here I dont know if it is right to say that this equals to $$ \sum_{n=1}^{\infty}\mathbb{E}(S_n) \mathbb{E}(N)=\sum_{n=1}^{\infty}\mathbb{E}\left(\sum_{k=1}^n X_k\right) \mathbb{E}(N)=\sum_{n=1}^{\infty}\sum_{k=1}^n\mathbb{E}\left(X_k\right) \mathbb{E}(N)=\mathbb{E}(X_1) \mathbb{E}(N) $$ I am not sure and I think I broke some math laws here. Any hints on this expectation and how to show that $S_N$ is a random variable? Thanks in advance!