Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 23H / c / Solution

Equip $\mathbb{N}$ with counting measure $\#$. For a nonnegative function $f:\mathbb{N}\to [0,\infty ]$, defineThis lies in $[0,\infty ]$ and agrees with the series as a Lebesgue integral against counting measure.

For $f:\mathbb{N}\to \mathbb{C}$, definewhen $f$ is integrable, which here meansThus the complex sum is defined under absolute convergence and can be obtained by integrating the real and imaginary parts.

A simple function on $\mathbb{N}$ is a function with finite range, equivalentlyfor a finite measurable partition by subsets $A_j\subseteq \mathbb{N}$. Its nonnegative sum is ${\sum }_j{a}_j\#A_j$. It is finite exactly when every level set with $a_j>0$ is finite, equivalently when $s$ has finite support after its zero level is discarded.