Is there an example of bounded non-measurable function?

Question

My teacher's lecture note states bounded function defined on a measurable set is not necessarily measurable. Can anyone help give a concrete example? Thank you!

Answer 1

Let $E\subset [0,1]$ be a non-measurable set. Then $$f(x)=\cases{1 & \text{ if $x\in E$}\\ 0 & \text{otherwise}}$$ is not measurable.

Answer 2

Suppose that $S$ is a set that is not measurable.

Then $1_S$ is not measurable.