While teaching statistics, I've grown sick of using terms such as "under (very) weak regulatory conditions..." and would like to be able to provide some solid examples of when these conditions are not met. For example, it's well known to any stats student that for a continuous random variable X and for a "sufficiently nice" function $g:\mathbb{R} \to \mathbb{R}$, we have E[$g$(X)]=$\int_{-\infty}^\infty g(x)f_X(x) \ \mathrm{dx}$. However, any stats student who knows their measure theory will know that, in this case, "sufficiently nice" means 'Borel Measurable'.
This leads me to the gist of my question. Are there any examples of non-Borel measurable functions that are simple enough to be explainable to someone who may have only just learned (or may not even know) set theory but are general enough to stress that the condition of 'g must be Borel measurable' is so weak that they won't really need to worry about it?