Consider the measure space $(\mathbb{N}, \mathcal{F}, \mu)$ where $\mu$ is the counting measure.
Consider a function $f:\mathbb{N}\rightarrow [0,\infty)$.
1) Is it true that, for any $f$, $$\int_{\mathbb{N}} f(k)d\mu$$ exists (finite or infinite)?
2) Is it true that, for any $f$, if $$\int_{\mathbb{N}} f(k)d\mu$$ exists (finite or infinite) [which is always the case if the answer to 1) is yes], then $f$ is measurable?