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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.)

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M. M. Rao also has a book, "Conditional Measures and Applications", in whch he discusses computation of conditional probabilities extensively, especially in the second edition. – Michael Greinecker Jan 5 '13 at 23:42
Thanks Michael. I haven't looked through the book yet, but was going to pick it up on Monday from the library. – R Hahn Jan 6 '13 at 0:00
In an applied situation I would imagine that you can't determine that $X = Y$ precisely due to measurement errors, so you should actually be conditioning on something like $|X - Y| < \epsilon$ and looking at the asymptotic behavior as $\epsilon \to 0$. It would make sense to look at $\frac{X}{Y}$ instead if the value of $\frac{X}{Y}$ is what I was directly measuring (rather than measuring $X$ and $Y$ separately). – Qiaochu Yuan Jan 6 '13 at 0:01
Qiaochu, your nice observation underscores my curiosity: why should our understanding of a physical problem depend on which of two measurements we make, when both measurements reflect the same physical state in the limit? Generically I do not expect an answer, but I am asking for actual examples where something concrete can be said. – R Hahn Jan 6 '13 at 0:29

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