Informatics Educational Institutions & Programs

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

It can be shown that the Lebesgue measure $\lambda$ on Euclidean space $\mathbb {R} ^{n}$ is locally finite, strictly positive, and translation-invariant. That is:

every point $x$ in $\mathbb {R} ^{n}$ has an open neighborhood $N_{x}$ with finite measure: $\lambda (N_{x})<+\infty ;$
every non-empty open subset $U$ of $\mathbb {R} ^{n}$ has positive measure: $\lambda (U)>0;$ and
if $A$ is any Lebesgue-measurable subset of $\mathbb {R} ^{n},$ and $h$ is a vector in $\mathbb {R} ^{n},$ then all translates of $A$ have the same measure: $\lambda (A+h)=\lambda (A).$

Geometrically, these three properties make the Lebesgue measure very useful. Although an infinite-dimensional space such as an $L^{p}$ 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 $G$ , there cannot exist a σ-finite, left-invariant Borel measure.^[1]

Non-Existence Theorem in Separable Banach spaces

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

Proof

Let $X$ be an infinite-dimensional, separable Banach space equipped with a locally finite translation-invariant measurement $\mu .$ To prove that $\mu$ is the trivial measure, it is sufficient and necessary to show that $\mu (X)=0.$

Like every separable metric space, $X$ is a Lindelöf space, which means that every open cover of $X$ has a countable subcover. It is therefore enough to show that there exists some open cover of $X$ by null sets, because by choosing a countable subcover, the σ-subadditivity of $\mu$ implies that $\mu (X)=0.$

Using local finiteness, suppose that for some $r>0,$ the open ball $B(r)$ of radius $r$ has a finite $\mu$ -measure. Since $X$ is infinite-dimensional, by Riesz's lemma there is an infinite sequence of pairwise disjoint open balls $B_{n}(r/4),$ $n\in \mathbb {N}$ , of radius $r/4,$ with all the smaller balls $B_{n}(r/4)$ contained within $B(r).$ 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 $\mu$ -measure zero.

Since $r$ was arbitrary, every open ball in $X$ has zero measure, and taking a cover of $X$ 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]}

References

^ 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.{{cite journal}}: 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.

[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.

[1]

[2]

[3]

Analysis in topological vector spaces
Basic concepts	Abstract Wiener space Classical Wiener space Bochner space Convex series Cylinder set measure Infinite-dimensional vector function Matrix calculus Vector calculus
Derivatives	Differentiable vector–valued functions from Euclidean space Differentiation in Fréchet spaces Fréchet derivative Total Functional derivative Gateaux derivative Directional Generalizations of the derivative Hadamard derivative Holomorphic Quasi-derivative
Measurability	Besov measure Cylinder set measure Canonical Gaussian Classical Wiener measure Measure like set functions infinite-dimensional Gaussian measure Projection-valued Vector Bochner / Weakly / Strongly measurable function Radonifying function
Integrals	Bochner Direct integral Dunford Gelfand–Pettis/Weak Regulated Paley–Wiener
Results	Cameron–Martin theorem Inverse function theorem Nash–Moser theorem Feldman–Hájek theorem No infinite-dimensional Lebesgue measure Sazonov's theorem Structure theorem for Gaussian measures
Related	Crinkled arc Covariance operator
Functional calculus	Borel functional calculus Continuous functional calculus Holomorphic functional calculus
Applications	Banach manifold (bundle) Convenient vector space Choquet theory Fréchet manifold Hilbert manifold