Solution
= Solution
Because every subset of $\mathbb N$ is open, the <Borel sigma-algebra of a discrete space> is
$$
\boxed{\mathcal B(\mathbb N)=\mathcal P(\mathbb N)}.
$$
If $f:\mathbb N\to\mathbb R$ or $f:\mathbb N\to[0,\infty]$ and $B$ is a Borel subset of the codomain, then $f^{-1}(B)$ is some subset of $\mathbb N$, hence is Borel. Thus every such function is a <Borel measurable function>.