I am working on my homework from Royden's Real Analysis:
Let $E$ be a measurable subset of $[a, b]$, and define $ f: [a, b] \to \mathbb{R} $ by $ f = \chi_E $. For each $ \epsilon > 0 $, show that there is a measurable subset $ E_0 \subseteq [a, b] $ and step-function $ h: [a, b] \to \mathbb{R} $ for which$$h = f \text{ on } E_0 \text{ and } m([a, b] \sim E_0) < \epsilon.$$
This is my solution (up to now):Since $E \subset [a,b]$, then $m(E) \leq b-a <\infty$. So for $\epsilon >0$, by a previous theorem from the book, there is a disjoint collection of open and bounded intervals $\lbrace I_k \rbrace_{k=1}^n$ for which if $U = \displaystyle \bigcup\limits_{k=1}^n I_k$, then$$ m(U\cap E)+m(U\cup E) = m(U\backslash E)+ m(E \backslash U) <\epsilon.$$Now, let $E_0 = \displaystyle \bigcup\limits_{k=1}^n I_k$ so $E_0$ is measurable. I let $h(x) = \displaystyle \sum\limits_{k=1}^n \chi_{I_k}(x)$. I want to show that $h=f$ on $E_0$ but I don't really know how. I mean by construction, then $E_0 \approx E$, but I don't know how to put them in words. Can you guys help me on this?