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 .

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