Let $(X, \mathcal E, \mu)$ be a $\sigma$-finite measure space. Prove:
A function $f: X \rightarrow \mathbb R^n$ is measurable if and only if there exists a sequence of simple, measurable functions $f_k: X \rightarrow \mathbb R^n$, such that $\mu(f^{-1}(\mathbb R^n \backslash \{0\})) < \infty$ and $f_k \rightarrow f$ for $k \rightarrow \infty$ almost everywhere.
I know that a function $f: X \rightarrow [0, \infty]$ is measurable if and only if there exists a sequence of monotonically increasing, measurable and simple functions $f_k: X \rightarrow [0, \infty]$, such that $f_k(x) \nearrow f(x), \forall x \in X$. I wanted to generalize this idea, but I haven't found a good sequence of functions yet. Can someone tell me if this approach has a chance of succeeding or is there an easier way?