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
.