Let $(X_i)_{i \in \mathbb{N}}$ be a sequence of integrable i.i.d. random variables on a probability space $( \Omega, \mathcal{F}, P )$ and let $N : \Omega \rightarrow \mathbb{N}_0$ be an integer-valued random variable (e.g. Poisson distributed) independent of $(X_i)_{i \in \mathbb{N}}$. Consider $$ S(\omega) = \sum_{i = 1}^{N(\omega)} X_i ( \omega), \quad \omega \in \Omega, $$ which is a random variable since for any $x \in \mathbb{R}$ we have $$ \{ S\leq x \} =\bigcup_{k=0}^{\infty} \left( \{ N=k \} \cap \left\{ \sum_{i=1}^{k} X_{i} \leq x \right\} \right). $$ Additionally, assume that $Y$ is also a random variable independent of $S$.
Does one have, for instance, $$E \left[ Y + \sum_{i = 1}^{N} X_i \Bigm| N = 1 \right] P(N=1) = E \left[ Y + X_1 \right] P(N=1)$$
Since \begin{align} E \left[ Y + \sum_{i = 1}^{N} X_i \Bigm| N = 1 \right] P(N=1) &= E \left[ 1_{\{ N = 1 \}} \left( Y + \sum_{i = 1}^{N} X_i \right) \right] \\ &= E\left[1_{\{ N = 1 \}} (Y + X_1)\right], \end{align} the question is, are $1_{\{N = 1\}}$ and $Y + X_1$ independent, provided that $Y$ and $S$ are independent?