To sum up: on the one hand, you tell me that defining probability measures on R^R is "nasty", "problematic"

But at the same time you allow yourself to believe that I don't enjoy sufficient background in mathematics to get a decent answer answer to my question.

Is that a joke?

Are you sure you teach mathematics???

I don't understand why you don't understand what a uniform probability distribution, proper or improper, is supposed to be??? Please explain