An infinite-dimensional Lebesgue measure is a type of measure defined on an infinite-dimensional normed vector space, more specifically a Banach space. It shares properties with the Lebesgue measure on finite-dimensional spaces.
However, the usual Lebesgue measure cannot be extended to all infinite-dimensional spaces. This limitation arises because any translation-invariant Borel measure on an infinite-dimensional separable Banach space is always either infinite for all sets or zero for all sets. Despite this, there are some instances of Lebesgue-like measures. These occur when the space is not separable, such as the Hilbert cube, or when one of the key properties of the Lebesgue measure is relaxed.
Motivation
[edit]The Lebesgue measure on the Euclidean space is locally finite, strictly positive, and translation-invariant. That is:
- every point in has an open neighborhood with finite measure:
- every non-empty open subset of has positive measure: and
- if is any Lebesgue-measurable subset of and is a vector in then all translates of have the same measure:
Motivated by their geometrical significance, constructing measures satisfying the above set properties for infinite-dimensional spaces such as the spaces or path spaces[disambiguation needed] is still an open and active area of research.
Statement of the theorem
[edit]On a non locally compact Polish group , there cannot exist a σ-finite and left-invariant Borel measure.[1]
This theorem implies that on an infinite dimensional separable Banach space - which can not be locally compact - a Lebesgue measure does not exist.
Non-Existence Theorem in Separable Banach spaces
[edit]Let be an infinite-dimensional, separable Banach space. Then, the only locally finite and translation invariant Borel measure on is a trivial measure. Equivalently, there is no locally finite, strictly positive, and translation invariant measure on .[1]
Proof
[edit]Let be an infinite-dimensional, separable Banach space equipped with a locally finite translation-invariant measurement To prove that is the trivial measure, it is sufficient and necessary to show that
Like every separable metric space, is a Lindelöf space, which means that every open cover of has a countable subcover. It is, therefore, enough to show that there exists some open cover of by null sets because by choosing a countable subcover, the σ-subadditivity of implies that
Using local finiteness, suppose that for some the open ball of radius has a finite -measure. Since is infinite-dimensional, by Riesz's lemma there is an infinite sequence of pairwise disjoint open balls , of radius with all the smaller balls contained within By translation invariance, all the smaller balls have the same measure, and since the sum of these measurements is finite, the smaller balls must all have -measure zero.
Since was arbitrary, every open ball in has zero measure, and taking a cover of which is the set of all open balls that completes the proof.
Nontrivial measures
[edit]Here are some examples of infinite-dimensional Lebesgue measures that can exist if the conditions of the above theorem are relaxed.
There are other measures that support entirely separable Banach spaces. One example is the abstract Wiener space construction, similar to products of Gaussian measures. Another approach is to consider a Lebesgue measurement of finite-dimensional subspaces within the larger space and look at 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.[citation needed]
See also
[edit]- Cylinder set measure – way to generate a measure over product spaces
- Cameron–Martin theorem – Theorem defining translation of Gaussian measures (Wiener measures) on Hilbert spaces.
- Feldman–Hájek theorem – Theory in probability theory
- Gaussian measure#Infinite-dimensional spaces – Type of Borel measure
- Structure theorem for Gaussian measures – Mathematical theorem
- Projection-valued measure – Mathematical operator-value measure of interest in quantum mechanics and functional analysis
- Set function – Function from sets to numbers
References
[edit]- ^ a b Oxtoby, John C. (1946). "Invariant measures in groups which are not locally compact". Trans. Amer. Math. Soc. 60: 216. doi:10.1090/S0002-9947-1946-0018188-5.
- ^ 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. arXiv:math/9210220. Bibcode:1992math.....10220H. doi:10.1090/S0273-0979-1992-00328-2. S2CID 17534021.
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: CS1 maint: multiple names: authors list (link) - ^ Oxtoby, John C.; Prasad, Vidhu S. (1978). "Homeomorphic Measures on the Hilbert Cube". Pacific J. Math. 77 (2): 483–497. doi:10.2140/pjm.1978.77.483.