I know that if we look at $\mathbb{R^n}$, a function $f:\mathbb{R^n} \rightarrow [0,\infty ]$ is lebesgue measurable if and only if there exisits a sequence of unsigned simpel functions ${f_n}:f:\mathbb{R^n} \rightarrow [0,\infty ]$ such that $\lim_{x\to\infty} f_n(x)=f(x)$ for all $x \in \mathbb{R^n} $
I am wondering if we have a similar result for abstract measure spaces $(X,\mathcal{B},\mu)$ or maybe for an abstract measure space that is complete? I think that would make sence but my book does noe mention it and i am not abel to prove it from my definiton of measurable functions in a abstract measure space which is that a function $f:X \rightarrow [0,\infty ]$ is measurable if and only if ${f^{-1}{(U)}}\in \mathcal{B}$ for all U $\subset[0,\infty]$ that are relatively open.