Let $f:\mathbb{N} \to [0,\infty ]$ defined as $f(i)=a_i$ for $i=1,2,3,....$,$a_i\geq 0$ and $\mathfrak{M}=\mathcal{P}(\mathbb{N})$ and $\mu$ the counting measure. It has then $$\int_{\mathbb{N}}f d\mu=\sum_{i=1}^{\infty}a_i$$
My idea would be to define a suitable sequence of simple functions such that $s_n \nearrow f$ to apply the MCT. How to define this sequence.? Any ideas thaks
Let $s_n(x) = \sum_{i = 1}^n a_i \chi_i(x)$, where $\chi_i(x) = 1$ if $x = i$ and $\chi_i(x) = 0$ if $x \neq i$. Since the $a_i$ are nonnegative, $s_n$ is an increasing sequence of nonnegative simple functions. Since $s_n(j) = a_j = f(j)$ for all $j \in \Bbb N$, $f = \lim_{n\to \infty} s_n$. Therefore
$$\int_{\Bbb N} f\, d\mu = \lim_{n\to \infty} \int_{\Bbb N} s_n \, d\mu = \lim_{n\to \infty} \sum_{i = 1}^n a_i \mu(i) = \lim_{n\to \infty} \sum_{i = 1}^n a_i = \sum_{i = 1}^\infty a_i.$$