Show that the following is not a Borel set in (this is the -algebra of functions over the domain unlike the previous onces we looked at where the domain was over the natural numbers).
If is a Borel set, then so is its complement . We know that all sets in must have this form for some and (proved in book and is simple). The function belongs to and therefore . Consider
Then, since the function belongs to . But clearly does not belong to . Hence is not a Borel set and neither is .