Let $(X, \mathcal{A},\mu)$ be a measure space and $f : X → [0,∞]$ be measurable. Show that there exist measurable sets $A_k, k = 1,2,...$ such that $$f = \sum_{k = 1}^{\infty} \frac{1}{k} \chi_{A_k}$$
Where $\chi_{A_k}(x) = 1$ if $x \in A_k$ and $0$ otherwise.
How do I even start to proceed with this problem? I'd be glad if anyone could point me to a relevant resource or gives a general direction for it.