# Hint needed on measure theory question: Approximation by semisimple functions

Note: A semisimple function takes on countably many values.

The hypotheses are that $f$ is measurable and real valued on a measurable set $E$. I would like to show that there is a sequence of semisimple functions $\{ f_n \}$ on $E$ that converge to $f$ uniformly on $E$.

I know that a real valued measurable function may be approximated pointwise by a sequence of simple functions on a measurable set $E$, with $$|\phi_n|<|f|$$ My idea is to somehow partition the set $E$ into a countable number of subsets, and then maybe tweak the the approximation theorem above to get uniform convergence? However, I am unsure of how to get uniform convergence out of this.

Any help getting a start would be appreciated. Thank you!

• What is a semisimple function? – Taufi Feb 15 '17 at 14:01

For positive integer $n$ define:$$f_n(x)=\frac1{n}\lfloor nf(x)\rfloor$$ Then $f_n$ is a semisimple measurable function with:$$f_n(x)\leq f(x)<f_n(x)+\frac1{n}$$
Hint 1. Fix $n\in{\mathbb N}$. For $k\in{\mathbb Z}$, let $E_{n,k}=\{\omega: f(\omega) \in [\frac{k}{2^n},\frac{k+1}{2^n})\}$.
If $\omega \in E_{n,k}$, then $0\le f(\omega) - k2^{-n}\le 2^{-n}$.