Any Borel function $f$, defined on some measurable Polish space with Borel $\sigma$-field $(E,\Sigma)$, and with non-negative, possibly infinite values admits a sequence of mb. simple functions which converges monotonely from below to $f$.
Is there an elegant way to find a sequence of non-negative Borel sets $(A_n) \in \Sigma^\mathbb N$ and non-negative real numbers $(b_n)$ such that $f=\sum_{n=0}^\infty b_n \mathbb 1_{A_n}$?
An answer for the case of finite-dimensional Euclidean space would be nice and sufficient already.
Let $s_{n}\uparrow f$, then $f=\displaystyle\sum_{n=1}^{\infty}(s_{n}-s_{n-1})$ with $s_{0}=0$, now exploit the nonnegative summand $s_{n}-s_{n-1}$ to a finite linear combination of characteristic functions of certain sets.