I'm not a logician. I knew I was going to get slated on this one but for some reason I wrote it anyway...

Sounds interesting. Just wondering if you hapened to post a preprint on the Arxiv?

I run the risk of incurring the wrath of some hardcore probabilists here, but for problems that are truly applied, I doubt even measure theory is that useful. As Hamming said, "Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether, say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly in that plane."

Presumably f(n) = 0 is not allowed...

That's interesting, but I'm not quite convinced - I can't see why translation invariance should fail from this example.

Semi-related question: mathoverflow.net/questions/59748/…