Talk:There is no infinite-dimensional Lebesgue measure

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[edit] re: unreferenced tag

What sources are needed exactly? the proof of the statement is given. 84.108.112.10 12:53, 3 February 2007 (UTC)

Obviously the theorem here is correct, as the proof shows. Giving a pointer to at least one textbook where this theorem has been published would be an aid to a reader who wants to see the theorem in context. A reference would also demonstrate more explicitly that it is not original research. CMummert · talk 13:51, 3 February 2007 (UTC)

[edit] comment by Kier07

I'm sorry -- could someone explain this proof to me? Why do we set c := B_r0/30(y)? What's the significance of 30? How do we know that B_r0/30(ei/2) is contained in B_r0(0) for all i? I don't even see why ei/2 is in B_r0(0) for all i. Doesn't this depend on how large r0 is? How do we know the balls are pairwise disjoint -- again, would that depend on how big r0 is? Should the proof conclude with, mu not equal to 0 implies mu(B_r0(0)) = infinity, but mu(B_r0(0)) < infinity by local finiteness, a contradiction? I'm really trying to follow this proof, because I find the result interesting, but I find that I'm hitting a brick wall. Thanks for any clarification! Kier07 06:09, 18 March 2007 (UTC)

I wish more readers would speak up when the proofs are too cryptic to understand. The proof currently on the page is quite terse. There was a typo that I will edit; it looks like ei/2 should be ei (r_0/2). Then it is clear that B_{r_0/30}(ei(r_0/2)) is contained in B_r0(0). The 30 is not uniquely chosen; it just needs to be small enough to allow the calculations to go through. I'll look into rewriting the proof. CMummert · talk 13:23, 18 March 2007 (UTC)

You're right about the typo. It was a simple matter of a missing r₀. 1/30 is a non-optimal choice of constant. I have done a partial re-write of the proof, adding some explanation and re-wording the contradiction at the end. Perhaps we can polish this together? Sullivan.t.j 13:41, 18 March 2007 (UTC)

I have a complete rewrite in my sandbox that I think is arranged better than the proof currently here. What do you think about it? CMummert · talk 13:48, 18 March 2007 (UTC)

I like it. I have made a few changes (hope you don't mind). I look forward to your revision, then we can post the updated version. Sullivan.t.j 14:12, 18 March 2007 (UTC)

Yes, please feel free to edit it. I want to look at it with fresh eyes in a few hours before making it live. CMummert · talk 14:27, 18 March 2007 (UTC)

[edit] the point

Would someone who is familiar with quantum physics add a paragraph explaining how this is related to the difficulty of formalizing certain integrals in quantum mechanics as Lebesgue integrals? I think this is the main real-life implication of the theorem presented here. CMummert · talk 13:59, 18 March 2007 (UTC)

We could do that. Are you referring to the fact that the naïve way of writing a path integral as an infinite iterated Lebesgue integral is not rigorous? In a way, that is what Wiener measure on path space tries to overcome. Sullivan.t.j 14:14, 18 March 2007 (UTC)

That sounds right. I am almost completely uninformed about quantum physics, so I am speaking here only of things that I have heard at talks. That's why I have to ask for someone else to write about it. CMummert · talk 14:26, 18 March 2007 (UTC)