The Borel-Kolmogorov paradox refers to situations where non-uniqueness in the notion of conditioning on a set of measure zero leads to apparent contradictions. As a formal matter, one requires instead to condition on "the" generating sigma algebra, which vanquishes non-uniqueness by fiat. For a technical explanations see this paper. Billingsley's measure theory book has a nice treatment as well.

I am looking for examples where a formal non-uniqueness was resolved by applied considerations which suggested a natural "tie-breaker".

A simple example from this paper illustrates the idea. Let $X$ and $Y$ be independent standard normal random variables. What is the conditional distribution of $X$ given that you are on the (measure-zero) line where $X = Y$? The answer will vary depending on if you condition on $Z_1 = 0$ where $Z_1 \equiv X - Y$ or $Z_2 = 1$ where $Z_2 \equiv X/Y$, to give just two of an infinite number of examples. So in a given situation which $Z$ is the "right" one to use?

My question is not about the Borel paradox or modeling random phenomena per se.

I am interested broadly in hearing about situations where

- we have a mathematically well defined condition ($x = y$ as above)
- we want to study some applied model when that condition is satisfied
- the conclusions we reach will differ depending on the way we approach (as taking a limit) that condition formally
Finally I am interested in how this ambiguity is resolved by "physical" considerations.

I would make the problem sharper if I could, but the reason I want examples is precisely to help focus my thinking. I find it intriguing that it is not enough to have a well defined condition and a well defined model, one must also justify (by way of interpretation) which limit to take!

I anticipate there are many examples from physics of which I am unaware and perhaps some from the literature on finite elements for solving PDEs.

(Apologies for the pay-wall links.)