Proof of the simple approximation lemma a) For the proof of the simple approximation lemma, our textbook says,

Let (c,d) be an open, bounded interval that contains the image of E, f(E), and $c=y_0 < y_1 < ... < y_n = d$ be a partition of the closed bounded interval [c,d] such that $y_k - y_{k-1} < \epsilon$ for $1 \leq k \leq n$. Define $I_k = [y_{k-1}, y_k)$ and $E_k = f^{-1}(I_k)$ for $ 1 \leq k \leq n$.
Since each $I_k$ is an interval and the function f is measurable, each set $E_k$ is measaurable.

I was a bit confused about this last sentence. I'm not sure what theorem they are using to say that $E_k$ is measurable because $I_k$ is an interval and f is measurable...

The definition of a function $f: A \to B$ being measurable is that for any measurable set $E \subseteq B$, $f^{-1}(E)$ is measurable, so this follows by definition.
