Let $f_n:\mathbb{R}\to\mathbb{R}$ for $n=1,2,...$. Then
\begin{equation}\sum_{n=1}^\infty f_n = \int_{\mathbb{N}}f_n\text{ d}\mu \end{equation} where $\mu$ is counting measure on $\mathbb{N}$.
I seem to not be able to prove this. Note that where are not summing values of $f$ at points in $\mathbb{N}$ but summing functions itself.
Thanks in advance!