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An infinite-dimensional Lebesgue measure (or Lebesgue-like measure) is a measure defined on an infinite-dimensional Banach space, which shares certain properties with the Lebesgue measure defined on finite-dimensional spaces.

The usual Lebesgue measure does not generalize to all infinite-dimensional spaces, because any translation-invariant Borel measure on an infinite-dimensional separable Banach space is either infinite on all sets, or zero on all sets. However, there are examples of Lebesgue-like measures when either the space is not separable (such as the Hilbert cube), or when one of the characteristic properties of the Lebesgue measure is relaxed.

Motivation

The Lebesgue measure on the Euclidean space is locally finite, strictly positive, and translation-invariant. That is:

Motivated by their geometrical signifance, constructing measures satisfying the above set properties, for infinite-dimensional spaces such as the spaces or path spaces is still an open and active area of research.

Statement of the theorem

On a non locally compact Polish group , there cannot exist a σ-finite, left-invariant Borel measure.[1]

Non-Existence Theorem in Separable Banach spaces

Let be an infinite-dimensional, separable Banach space. Then the only locally finite and translation invariant Borel measure on is the trivial measure. Equivalently, there is no locally-finite, strictly positive, translation invariant measure on .

Proof

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 completes the proof.

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.[citation needed]

See also

References

  1. ^ 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.
  2. ^ 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.((cite journal)): CS1 maint: multiple names: authors list (link)
  3. ^ 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.