Suppose that we have an i.i.d. sequence of random varaible $(X_n)_{n\in \mathbb{N}}$, $M$ is a stopping time for $S_n = \sum_{i=1}^n X_i$, and that all the assumption for the validity of Wald's equations are verified so that
$\mathbb{E}\left\{\sum_{i=1}^M X_n\right\} = \mathbb{E}\left\{M\right\}\mathbb{E}\left\{X\right\}$.
If $f$ is a continuous, deterministic function so that $f(X)$ is always limited, the following
$\mathbb{E}\left\{\sum_{i=1}^M f(X_n)\right\} = \mathbb{E}\left\{M\right\}\mathbb{E}\left\{f(X)\right\}$
is valid ??