Let \((\Omega, \mathcal{A})\) be a measurable space and \(f\colon \Omega \to [0,\infty]\) measurable . Then a sequence of simple functions \((f_n)_{n\in \mathbb{N}}\) exists with \(f_n \nearrow f\) point wise. Especially, we there are \(A_1, A_2, \ldots \in \mathcal{A}\) and \(\alpha_1,\alpha_2,\ldots \ge 0\) such that

\begin{equation*} f=\sum_{n=1}^{\infty} \alpha_n 𝟙_{A_n}. \end{equation*}[1, Satz 1.96]

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  1. A. Klenke, Wahrscheinlichkeitstheorie. Berlin, Heidelberg: Springer Berlin Heidelberg, 2020. doi:10.1007/978-3-662-62089-2