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There are no Lebesgue measure on infinite dimensional Banach space. However, there is a Lebesgue measure on the Hilbert cube $[0,1]^\mathbb{N}$. What are the differences between this measure and the finite-dimensional Lebesgue measure on $[0,1]^n$?

user88163
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  • Can you define or link to this measure? I guess you would consider the volume of an $\mathbb{N}$-cube (that is, a cube with extent in every dimension) as an infinite product, then do the usual dance with coverings by these cubes? – Andrew Poelstra Jul 29 '13 at 15:44
  • http://math.stackexchange.com/a/269369 http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.pjm/1102806462 – user88163 Jul 29 '13 at 16:01

3 Answers3

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Here is a difference: consider the set $D_t=\{x\in S\mid m(x)\ \text{exists},\ \frac12-t\leqslant m(x)\leqslant\frac12+t\}$ where $S=[0,1]^n$ or $[0,1]^\mathbb N$, $m(x)=\frac1n\sum\limits_{k=1}^nx_k$ if $x$ is in $[0,1]^n$, and $m(x)=\lim\limits_{n\to\infty}\frac1n\sum\limits_{k=1}^nx_k$ if $x$ is in $[0,1]^\mathbb N$ and the limit exists. Then, $\mathrm{Leb}(D_t)\to0$ when $t\to0$ if $S=[0,1]^n$, while $\mathrm{Leb}(D_t)=1$ for every positive $t$ if $S=[0,1]^\mathbb N$.

Did
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Corollary 0.8 in Folland's Intro to Partial Differential Equations is that the volume of the unit ball in $\mathbb{R}^n$ is given by $$\omega_n = \frac{2\pi^{n/2}}{\Gamma(n/2)},$$ which tends to 0 as $n\to\infty$. This implies that the volume of any $\mathbb{N}$-sphere will be zero, which probably has some weird implications, e.g. you cannot cover any set of positive measure with countably many $\mathbb{N}$-spheres.

Andrew Poelstra
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    You should consider the precise meaning of the term "$\mathbb N$-sphere" that you use here. For one thing, Hilbert cube is contained in a ball of the Hilbert space, when it's realized inside of a Hilbert space. – 40 votes Jul 29 '13 at 18:01
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Lebesgue measure in $[0,1]^{N}$ is the product of infinitely many copies of linear Lebesgue measures in $[0,1]$ unlike the Lebesgue measure in $[0,1]^n$ which is the product of $n$ copies of linear Lebesgue measures in $[0,1]$.

George
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