## Integrable

Let $\mu$ be a measure on the natural numbers s.t $\mu(\left\{n\right\})=\alpha_n$ for all $n\in \mathbb{N}$

Show that a function $f:\mathbb{N}\to\mathbb{R}$ is $\mu$-integrable $\Leftrightarrow

\sum_{n\in\mathbb{N}}|f(n)|\alpha_n$
is convergent.

What is the value of $\int fd\mu$?

I don't quite understand what's to be shown exactly. According to my definitions f is integrable when f is measurable and $\int_{\mathbb{N}}|f|d\mu<\infty$

But isn't the integral equal to $\sum_{n\in \mathbb{N}}|f(n)|d\mu(\left\{n\right\}) = \sum_{n\in \mathbb{N}}|f(n)|\alpha_n....<\infty$?

Isn't $\Rightarrow$ immediate then?

But then what's to be shown for $\Leftarrow$. A little guidance is greatly appreciated