Let $\mu : {\mathcal{P}} $($\mathbb{N}$) $\to$ [$0$,$\infty$] be the counting measure on the measurable space ($\mathbb{N}$,$\mathcal{P}$($\mathbb{N}$)) and let
$$ f:\mathbb{N} \to [0,\infty] $$ be an illustration.
Show that that $f \in M_{+}(\mathbb{N}, \mathcal{P}(\mathbb{N}))$ and compute $$ \int_{\mathbb{N}} f d\mu $$
I will assume that $f \in M_+(\mathbb{N},\mathcal{P}(\mathbb{N}))$ means that f is measurable in that measure space of the power set of the natural numbers. What is an ilustration function?
Is $f: (\mathbb{N},\mathcal{P}(\mathbb{N})) \to (\mathbb{R}_{\infty},\mathcal{B}(\mathbb{R}_{\infty}))$ ??
If so, then $f$ is measurable if $f^{-1}(B) \in \mathcal{P}(\mathbb{N})$ whenever $B \in (\mathbb{R}_{\infty},\mathcal{B}(\mathbb{R}_{\infty}))$
Also I from what you write I can guess that
$\int_{N}f d\mu = \infty$.