Английская Википедия:Infinite-dimensional Lebesgue measure
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The Lebesgue measure is a tool used in functional analysis and measure theory to give a notion of 'volume' to subsets of <math>\mathbb{R}^n</math>. The problem of extending the Lebesgue measure to an analogous measure on an infinite dimensional space is trivial in the case of separable Banach spaces: any translation invariant Borel measure on an infinite dimensional separable Banach space is either infinite on all open sets, or the zero measure.
Numerous examples of such measures exist for non-separable Banach spaces or non-translation invariant measures. Lebesgue measure can be extended to various subsets of infinite-dimensional space, like the Hilbert cube.
Motivation
It can be shown that the Lebesgue measure <math>\lambda</math> on Euclidean space <math>\Reals^n</math> is locally finite, strictly positive, and translation-invariant. That is:
- every point <math>x</math> in <math>\Reals^n</math> has an open neighbourhood <math>N_x</math> with finite measure <math>\lambda(N_x) < + \infty;</math>
- every non-empty open subset <math>U</math> of <math>\Reals^n</math> has positive a measure <math>\lambda(U) > 0;</math> and
- if <math>A</math> is any Lebesgue-measurable subset of <math>\Reals^n,</math> <math>T_n : \Reals^n \to \Reals^n,</math> <math>T_h(x) = x + h,</math> denotes the translation map, and <math>(T_h)_*(\lambda)</math> denotes the push forward, then <math>(T_h)_*(\lambda)(A) = \lambda(A).</math>
Geometrically, these three properties make the Lebesgue measure very useful. Although an infinite-dimensional space such as an <math>L^p</math> space or the space of continuous paths in Euclidean space would be clean to have a similar measurement to work with, it is not yet be proven to be possible.
Statement of the theorem
On a non locally compact Polish group <math>G</math>, there cannot exist a σ-finite, left-invariant Borel measure.[1]
Separable Banach spaces
Theorem: Let <math>(X, \|\cdot\|)</math> be an infinite-dimensional, separable Banach space. Then the only locally finite and translation-invariant Borel measure <math>\mu</math> on <math>X</math> is the trivial measure, with <math>\mu(A) = 0</math> for every measurable set <math>A.</math> Equivalently, every translation-invariant measure that is not identically zero assigns an infinite measurement to all open subsets of <math>X.</math>
Proof of the Theorem
Let <math>X</math> be an infinite-dimension, separable Banach space equipped with a locally finite translation-invariant measurement <math>\mu.</math>
Like every separable metric space, <math>X</math> is a Lindelöf space, which means that every open cover of <math>X</math> has a countable subcover.
To prove that <math>\mu</math> is the trivial measurement, it is sufficient and necessary to show that <math>\mu(X) = 0.</math> To prove this, show that there exists some non-empty open sets of <math>N</math> that measure zero, because then <math>\{x + N: x \in X\}</math> will be an open cover of <math>X</math> by sets of the measurement <math>\mu(x + N) = \mu(N) = 0</math> (by translation-invariance); after picking any countable subcover of <math>X</math> by these measurement zero sets, <math>\mu(X) = 0</math> will follow from the σ-subadditivity of <math>\mu.</math>
Using local finiteness, suppose that for some <math>r > 0,</math> the open ball <math>B(r)</math> of radius <math>r</math> has a finite <math>\mu</math>-measurement. Since <math>X</math> is defined as being infinite-dimensional by Riesz's lemma there is an infinite sequence of pairwise disjoint open balls <math>B_n(r/4),</math> <math>n \in \N,</math> of radius <math>r/4,</math> with all the smaller balls <math>B_n(r/4)</math> contained within the larger ball <math>B(r).</math> By translation-invariance, all the smaller balls have the same measurement; since the sum of these measurements is finite, the smaller balls must all have <math>\mu</math>-measurement of zero.
Nontrivial measures
The following are examples where a notion of an infinite-dimensional Lebesgue measure exists, once the conditions of the above theorem are loosened.
There are other kinds of measures that support entirely separable Banach spaces such as the abstract Wiener space construction, which gives the analog of products of Gaussian measures. Alternatively, one may consider a Lebesgue measurement of finite-dimensional subspaces on the larger space and consider the so-called prevalent and shy sets.[2]
The Hilbert cube carries the product Lebesgue measure[3] and the compact topological group given by the Tychonoff product of an infinite number of copies of the circle group which is infinite-dimensional and carries a Haar measure that is translation-invariant. These two spaces can be mapped onto each other in a measure-preserving way by unwrapping the circles into intervals. The infinite product of the additive real numbers has the analogous product Haar measure, which is precisely the infinite-dimensional analog of the Lebesgue measure.Шаблон:Citation needed
See also
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References
Шаблон:Analysis in topological vector spaces Шаблон:Measure theory Шаблон:Functional analysis
