Let $X_1,X_2,\ldots$ be i.i.d. random variables, and $S_n=X_1+\cdots+X_n$. Assume that $0 < \mathbf{E}(X_1) < \infty$ (but don't assume that the $X_i$ are $>0$). Let $N$ be the almost surely finite stopping time $N=\inf \{ n ; S_n>0 \}$. Let $A \subset \mathbf R$ be some Borel set. We are interested in $$ U = \mathbf{E} \left[ \sum_{n \geq N} \mathbf{1}_A (S_n) \right] = \mathbf{E} \left[ \sum_{n \geq N} \mathbf{1}_{A-S_N} (S_n-S_N) \right] $$
Now it is tempting to say that $S_n-S_N$ has the same distribution as $S_{n-N}$ when $n \geq N$, so that if we let, for any real $a$, $\mu(a)=\mathbf{E}\left[ \sum_{n \geq 0} \mathbf{1}_{A-a} (S_n) \right]$, we have in fact $$ U = \mathbf{E} [\mu(S_N)]$$
I did not succeed in making this derivation formal. Could you help me? I'm really stuck on this.