Suppose we have a set $S$ (as opposed to a sequence) of non-negative real-valued functions with
$$\hat{s}(x) := \sup \{s(x) ~|~ s \in S\}~.$$
Does
$$\sup \left\{\int s(x) ~ d \mu ~\middle|~ s \in S \right\} ~{}={}~ \int \hat{s} ~d \mu$$
hold for the Lebesgue integral, where $\mu$ is a probability measure with countable support?
It does not hold.
For $c \in [0,1]$ consider $\chi_{\{c\}}$, the indicator function of the set $\{c\}$.
$\chi_{\{c\}}$ is measurable since $\{c\}$ is measurable, and we have:
$$\int_{\mathbb{R}} \chi_{\{c\}} \,d\lambda= \lambda(\{c\}) = 0$$
The pointwise supremum of the set of functions $\left\{\chi_{\{c\}} : c \in [0,1]\right\}$ is the function $\chi_{[0,1]}$.
We have:
$$\int_{\mathbb{R}} \chi_{[0,1]} \,d\lambda= \lambda([0,1]) = 1$$