Given a measure space $(\Bbb R, \mathcal P(\Bbb R), \mu_\Bbb N)$ where $\mathcal P(\Bbb R)$ denotes the power set of $\Bbb R$ and $\mu_\Bbb N$ is defined by $\mu_\Bbb N(A)= \vert {A \cap \Bbb N}\vert$= quantity of natural numbers in $A$,
So far it is proven that $f,g:\Bbb R \to \mathbb R$ coincide on $\mathbb N$ iff $f(x)=g(x)$ $\mu_\mathbb N$-almost everywhere and that $f:\Bbb R \to \Bbb R$ is $\mu_\Bbb N$-integrable iff $\sum_{n=1}^{\infty} f(n)$ absolutely converges , where $\int f d\mu_\Bbb N$ =$\sum_{n=1}^{\infty} f(n)$.
The last thing I need to prove now is to deduce the comparison test for series from the dominated convergence theorem. Any ideas on how to do so?