We are given an enumeration of the set $\Lambda_X = \{y_i \mid i \in \mathbb{N}\}$. Show that for all $x \in \mathbb{R}$, $$f_n(x) = \sum_{k=0}^{n} \sum_{i=0}^{+\infty} \mathbb{P}\left(S_k = y_i\right) g\left(x - y_i\right)$$
We are given an enumeration of the set $\Lambda_X = \{y_i \mid i \in \mathbb{N}\}$. Show that for all $x \in \mathbb{R}$,
$$f_n(x) = \sum_{k=0}^{n} \sum_{i=0}^{+\infty} \mathbb{P}\left(S_k = y_i\right) g\left(x - y_i\right)$$