Nonmeasurable Functions

Question

Reference

This question is related to: Banach Spaces: Uniform Integral vs. Riemann Integral

Problem

What are examples of real-valued functions:

Bounded & Non-Step & Non-Measurable

(Especially, it should be not a.e. a step!)

Answer 1

It is still quite trivial but at least...

Given the Lebesgue measure $\lambda:[0,1]\to(0,\infty)$.

Consider a Vitali set $\mu_*(V)<\mu^*(V)$.

Construct the function: $$f:[0,1]\to\mathbb{R}:f(x):=x^2\chi_V(x)$$ This one is not sum of measurable plus step but product.

Answer 2

Let $V$ be a non-measurable set, it must be uncountable, select a countable points from it $\{x_i\}$, define $f=\chi_V$ except those points, and let $f(x_i)=\frac{1}{n}$.

So $f$ is bounded by $1$, unmeasurable, has countable many values.