Talk:Infinite-dimensional Lebesgue measure

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)

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)

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)

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)

Article contradicted itself

The claim "There is no analog of Lebesgue measure on Banach spaces" is absolutely false, there are easy counterexamples, like the Haar measure on the infinite product of the circle group mentioned *two paragraphs down*. The correct claim is that there is no sigma finite measure or a nontrivial measure on a separable Banach space. These claims are correct. The infinite product Lebesgue measure is a perfectly normal construction. — Preceding unsigned comment added by 31.168.158.220 (talk) 15:35, 9 January 2023 (UTC)

The statement "There is no analog of Lebesgue measure on Banach spaces" is not falsifiable; it depends on what one wants in an 'analog.' Some might want a translation-invariant, sigma finite measure that gives non-zero measure to open balls. This, you concede, does not exist and so there is no analog, in this sense. I think the folklore statement is useful to people trying to learn this (me) in that it quickly communicates that one cannot transport the intuition of Lebesgue measure to larger spaces. Moreover, from what I can gather, there remain important problems for the application of shy sets on an infinite dimensional Banach space: for *every* probability measure, there is a shy set that gets full probability (see abstract of Stinchcombe, Proc. of AMS 2001, 451-457. I have not yet studied the paper in detail). Thus, for talking about probability at least, it seems that shyness lacks the (applied) power that 'Lebesgue null' has on finite dimensions, adding credence to the folklore. 2A02:AB88:5987:A700:B4F3:8B4B:EE22:B38B (talk) 10:00, 27 March 2023 (UTC)

My reading of OP's comment is that they agree with you on the meaning of "analog". They seem to be arguing about missing "separable" off the list of conditions. Danielittlewood (talk) 18:32, 30 March 2024 (UTC)

There are infinite-dimensional Lebesgue measures in an arbitrary infinite-dimensional separable Banach space

An existence of the measure $\lambda$, which can be considered as an infinite analog of the Lebesgue measure on an infinite-dimensional topological vector space $R^{\infty}$, has been proved in [1]. Some generalizations of the Baker measure $\lambda$ have been constructed in [2]. Let $B$ be an arbitrary infinite-dimensional separable Banach space $B$ with an absolutely convergent Schauder basis ( Note here that an existence of Schauder basis in $B$ implies an existence of an absolutely convergent Schauder basis ). A natural embedding of $B$ into $R^{\infty}$ allows us to construct a measure $\mu$ which can be considered now as an infinite analog of the Lebesgue measure on $B$. It is of some interest that $\mu$ is not $\sigma$-finite but if a set $X \subset B$ is of $\mu$ -measure zero then $X$ is shy in the sense of [3]. Such measures are called generators of shy sets (cf. [4]).

A notion of generators of shy sets in Polish topological vector spaces firstly has been introduced in [4] and have been considered their various interesting applications. In particular, here has been demonstrated that this class contains specific measures which naturally generate early implicitly introduced classes of null sets (cf. [4]). For example, here has been constructed : 1) Mankiewicz generator which generates exactly the class of all "cube null" sets; 2) Preiss -Tiser generators which generates exactly the class of all "Preiss -Tiser null" sets , etc. Moreover, such measures (unlike $\sigma$-finite Borel measures) possess many interesting, some-times unexpected, geometric properties

Motivation. It can be shown that Lebesgue measure $λ^n$ on Euclidean space $R^n$ has the following properties:

a) for every point $x$ in $R^n$ there is a parallelepiped $P_x$ with the center at $x$ such that $0< λ^n(P_x) < +\infty$;

b) every non-empty open subset $U \subseteq R^n$ has positive measure, i.e., $λ^n(U) > 0$;

c) if $A$ is a Lebesgue-measurable subset of $R^n$ and for $h \in R^n$ a shift operator $T_h : R^n → R^n$ is defined by $T_h(x) = x + h (x \in R^n)$ , then $(λ^n)(T_h(A)) = λ^n(A)$.

d) any set of $λ^n$-measure zero is shy in the sence of [2].

Geometrically speaking, these properties make Lebesgue measure very nice to work with. If we consider an arbitrary infinite-dimensional separable Banach space then such a Borel measure always exists(see, for example, [5]).

19:42, 13 May 2012‎ 78.139.167.160

References [1] Baker, Richard L. "Lebesgue measure on $R^[\infty}$. II. Proc. Amer. Math. Soc. 132 (2004), no. 9, 2577—2591.

[2] G.Pantsulaia, On ordinary and standard products of infinite family of $\sigma$ -finite measures and some of their applications. Acta Math. Sin. (Engl. Ser.) 27 (2011), no. 3, 477--496.

[3] Hunt, Brian R. and Sauer, Tim and Yorke, James A. (1992). "Prevalence: a translation-invariant "almost every" on infinite-dimensional spaces". Bull. Amer. Math. Soc. (N.S.) 27 (2): 217–238. doi:10.1090/S0273-0979-1992-00328-2. (See section 1: Introduction)

[4] G.Pantsulaia, On generators of shy sets on Polish topological vector spaces, New York J. Math., 14 ( 2008) , 235 – 261.

[5] Gill, Tepper; Kirtadze, Aleks; Pantsulaia, Gogi; Plichko, Anatolij. Existence and uniqueness of translation invariant measures in separable Banach spaces.Funct. Approx. Comment. Math. 50 (2014), no. 2, 401--419.

But the measure constructed by Baker ([1]) is not σ-finite, hence it is not a 100% analogue of the Lebesgue measure.--Tensorproduct (talk) 20:52, 15 March 2024 (UTC)

Re-work the Introduction

This is much in the same line of thought as the heading above "Article contradicted itself". The point of mathematics is its precise and rigorous logical structure. To claim something is "folklore" or that it is "false" without due regard for the logical conditions required to make the statement true is wholly inaccurate. Indeed, there is no counter example to the triple conditions of translation-invariant, separable, and sigma-finite.

As a consequence, it is necessary to re-write the introduction so that it states the no-go theorem properly and fairly. But also, it should state the myriad generalizations that can dispense with one or more of the triple conditions (but not all 3). Currently, the introduction is too chaotic and does not read in the tone appropriate for an encyclopedia.

This is an editor's call to re-work the introduction so that it uses the classical theorem in an accurate way as a point of departure to discuss the myriad interesting generalizations featured in this article, and thus justifying the title of the article. The article can then serve as a place where readers can find a host of different notions of generalization of Lebesgue measure on infinite dimensional spaces without the unnecessary disparaging of classical results that are clearly true. MMmpds (talk) 19:33, 8 August 2023 (UTC)

I added the statement of the real theorem, which is that on "non locally compact Polish group" an analog of the Lebesgue measure does not exist. I hope this finally solves everything.--Tensorproduct (talk) 20:54, 15 March 2024 (UTC)

Hi Tensorproduct, I found it hard to parse your theorem statement in the surrounding context of the article. This might be something obvious to experts, but it isn't clear to me how the theorem about non-locally-compact Polish groups is related to the theorem for separable Banach spaces that is proved in the article. Is every separable Banach space a non-locally-compact Polish group? Is it a strict generalisation, or just a different way of stating the same thing? Are there other interesting examples that are different from Banach spaces with addition as the group operation? Danielittlewood (talk) 18:07, 30 March 2024 (UTC)

I rephrased the introduction. I wonder if it is more inline with what you have in mind? I hope I did not introduce any mistakes. It didn't look to me like the version of the intro had any technical errors, but I agree with your comments about tone. Danielittlewood (talk) 15:56, 30 March 2024 (UTC)