I want to know If my resolution is correct.
We have to prove that, being $X = \Bbb{N}$, $\mu$ the counting measure and $ f(n) = \frac{1}{n}$, $f \notin L_1$.
Let $E_n$ = {$n$}. Then, $\cup^{\infty}_{n=1}E_n = \Bbb{N}$ and $\mu(E_n) = 1$ for all n. Also, these sets are disjoint.
Now, $\int_{\Bbb{N}} \vert f \vert d\mu$ = $\sum^{\infty}_{n= 1} \int_{{E_n}} \vert f \vert d\mu $ = $ \sum^{\infty}_{n=1} \frac{1}{n} \mu(E_n) $ = $ \sum^{\infty}_{n=1} \frac{1}{n}$ and this is the harmonic series, does not converge and goes to infinity.
Its the resolution correct?