Measurable if and only if absolutely convergent

Question

Let $N$ be the set of natural numbers, $M = 2^N$, and $c$ the counting measure defined by setting $c(E)$ equal to the number of points in $E$ if $E$ is finite and $\infty$ if $E$ is an infinite set. Prove that every function $f: N \rightarrow R$ is measurable with respect to $c$ and that $f$ is integrable over $N$ with respect to $c$ if and only if the series $\sum^\infty _ {k=1} f(k)$ is absolutely convergent in which case $\int_N fdc = \sum^\infty _{k=1} f(k)$.

Answer 1

Note that the only set of zero-measure is the emptyset.

If $f\colon\Bbb N\to \Bbb R_{\geqslant 0}$ is a function, note that $\sum_{k=1}^nf(k)\chi_{\{k\}}$ is a simple function smaller than $f$. Now go back to the definition of integrability of a non-negative function.