Given a family of measurable functions with , shall we have measurable? If each is continuous, then is lower semicontinuous, so is measurable. However, it may not be true in general.
For simplicity, we take as the domain of all functions below. Let be Borel -algebra and be its completion with respect to its usual metric on . We say a function is measurable if for any , and denoted by .
Let’s take be a non-Lebesgue-measurable set and functions be given by
It is trivial to show that . Indeed, one can show this by
It is also interesting to show that, if is given, then the corresponding .