Talk:There is no infinite-dimensional Lebesgue measure

From Wikipedia, the free encyclopedia

Contents 1 re: unreferenced tag 2 comment by Kier07 3 the point 4 Banach space?

[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)

[edit] Banach space?

Is this theorem true for Banach spaces too? If so, shall we mention that? Temur 20:59, 3 August 2007 (UTC)

Good question. The "usual" strictly positive, locally finite measure on a separable Banach space E is abstract Wiener measure, which is only translation quasi-invariant (by the Cameron-Martin theorem) so one would suspect that the theorem holds for Banach spaces. Looking at the proof given in the article, and the similar one that I have in my undergraduate course notes, it seems that the first step (showing that μ(E) > 0) goes through for any Banach space E. In the case that E is separable, we are also free to pick a countable basis for E, but have no notion of orthonormality, so no Pythagoras theorem. Hence, the proof for Banach spaces will need some additional trickery, even if it is true. Sullivan.t.j 22:36, 4 August 2007 (UTC)

I found the following in Cameron-Martin theorem: "If E is a separable Banach space and μ is a locally finite Borel measure on E that is equivalent to its own push forward under any translation, then either E has finite dimension or μ is the trivial (zero) measure." Does this result imply that the theorem is true for Banach spaces? Temur 19:37, 6 August 2007 (UTC)

Never mind. I did not notice you already included Banach space case. Temur 19:39, 6 August 2007 (UTC)

The theorem that you mentioned is in some sense "even worse" than the one in this article, since it tells us that not only is the search for "decent" translation-invariant measures on infinite-dimensional Banach spaces hopeless, the search for translation-quadi-invariant ones is hopeless, too! Sullivan.t.j 09:51, 7 August 2007 (UTC)

I think the proof in the article works for nonseparable case as well since it does not explicitly specify how to choose those small balls. By the way, is not the Wiener measure translation quasi-invariant? Temur 16:52, 8 August 2007 (UTC)

The new proof (the one added by me approx. two days ago) handles the non-separable case in that it shows that some open sets will have zero measure, even though the measure might not actually be the trivial one. So, in the non-separable case, you lose strict positivity; in the separable case, you lose everything! As for quasi-invariance of Wiener measure on an abstract Wiener space, the Cameron-Martin formula only gives quasi-invariance for translation by so-called "Cameron-Martin directions", i.e. elements of i(H), a proper subset of E (see the AWS article for the notation). This limitation makes sense, because if Wiener measure were quasi-invariant under all translations of an infinite-dimensional space E, it would have to be the trivial measure. In summary, an abstract Wiener measure is a strictly positive, locally finite, Gaussian Borel measure that is quasi-invariant only under translations by Cameron-Martin directions. Sullivan.t.j 18:16, 8 August 2007 (UTC)