I am having trouble computing integration w.r.t. counting measure. Let $(\mathbb{N},\scr{P}(\mathbb{N}),\mu)$ be a measure space where $\mu$ is counting measure. Let $f:\mathbb{N}\rightarrow{\mathbb{R}}$ be a non-negative bounded measurable function. Then, what is $\int_{\mathbb{N}}fd\mu$? What's gonna happen if we remove the boundedness in $f$, i.e. just let $f$ be an arbitrary non-negative measurable function?
What happens if we relace $\mathbb{N}$ by a general set $X$?