I want to find a sequence such that is defined on (0,1) for all natural numbers , we have (m=Lebesgue measure), and does not exist for any x in (0,1).

My initial idea, which I know does not work, was to define something like this. For n odd, let be for irrational x, and 0 otherwise. For n even, let be 0 for irrational x and 1 for rational x. That satisfies the first condition about the limit of the integral, but does not satisfy the second condition since the limit of is 0 on the irrationals.

Even though I know this doesn't work, I was wondering if someone here would know of a way I could think about tweaking this to work, or am I barking up the wrong tree, so to speak?