Infinite-dimensional Lebesgue measure

In functional analysis and measure theory, there is a folklore claim that there is no analog of the Lebesgue measure on an infinite-dimensional Banach space. The claim states that there is no translation of invariant measure on a separable Banach space—because if any ball has a non-zero, non-infinite volume; a slightly smaller ball has zero volume and an assessable number of similar smaller balls covering the space. However, this folklore statement is entirely false. The surveyable product of Lebesgue measure's translation is invariant and gives the notion of volume as the infinite product of lengths. Only the domain on which this product measure is defined must necessarily be non-separable, but the measure itself is not sigma finite.

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.

The Hilbert cube carries the product Lebesgue measure 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.

Motivation

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

• every point ${\displaystyle x}$  in ${\displaystyle \mathbb {R} ^{n}}$  has an open neighbourhood ${\displaystyle N_{x}}$  with finite measure ${\displaystyle \lambda (N_{x})<+\infty ;}$ 
• every non-empty open subset ${\displaystyle U}$  of ${\displaystyle \mathbb {R} ^{n}}$  has positive a measure ${\displaystyle \lambda (U)>0;}$  and
• if ${\displaystyle A}$  is any Lebesgue-measurable subset of ${\displaystyle \mathbb {R} ^{n},}$  ${\displaystyle T_{n}:\mathbb {R} ^{n}\to \mathbb {R} ^{n},}$  ${\displaystyle T_{h}(x)=x+h,}$  denotes the translation map, and ${\displaystyle (T_{h})_{*}(\lambda )}$  denotes the push forward, then ${\displaystyle (T_{h})_{*}(\lambda )(A)=\lambda (A).}$ 

Geometrically speaking, these three properties make the Lebesgue measure elegant. When we consider an infinite-dimensional space such as an ${\displaystyle L^{p}}$  space or the space of continuous paths in Euclidean space, it would be clean to have a similar measurement to work with; however, this is not possible.

Statement of the theorem

Let ${\displaystyle (X,\|\cdot \|)}$  be an infinite-dimensional, separable Banach space. Then the only locally finite and translation-invariant Borel measure ${\displaystyle \mu }$  on ${\displaystyle X}$  is the trivial measure, with ${\displaystyle \mu (A)=0}$  for every measurable set ${\displaystyle A.}$  Equivalently, every translation-invariant measure that is not identically zero assigns an infinite measurement to all open subsets of ${\displaystyle X.}$ 

Proof of the theorem

Let ${\displaystyle X}$  be an infinite-dimension, separable Banach space equipped with a locally finite translation-invariant measurement ${\displaystyle \mu .}$ 

Like every separable metric space, ${\displaystyle X}$  is a Lindelöf space, which means that every open cover of ${\displaystyle X}$  has a countable subcover.

To prove that ${\displaystyle \mu }$  is the trivial measurement, it is sufficient and necessary to show that ${\displaystyle \mu (X)=0.}$  To prove this show that there exist some non-empty open sets of ${\displaystyle N}$  that measure zero because then ${\displaystyle \{x+N:x\in X\}}$  will be an open cover of ${\displaystyle X}$  by sets of the measurement ${\displaystyle \mu (x+N)=\mu (N)=0}$  (by translation-invariance); after picking any countable subcover of ${\displaystyle X}$  by these measurement zero sets, ${\displaystyle \mu (X)=0}$  will follow from the σ-subadditivity of ${\displaystyle \mu .}$ 

Using local finiteness, suppose that for some ${\displaystyle r>0,}$  the open ball ${\displaystyle B(r)}$  of radius ${\displaystyle r}$  has a finite ${\displaystyle \mu }$ -measurement. Since ${\displaystyle X}$  is defined as being infinite-dimensional by Riesz's lemma there is an infinite sequence of pairwise disjoint open balls ${\displaystyle B_{n}(r/4),}$  ${\displaystyle n\in \mathbb {N} ,}$  of radius ${\displaystyle r/4,}$  with all the smaller balls ${\displaystyle B_{n}(r/4)}$  contained within the larger ball ${\displaystyle B(r).}$  By translation-invariance, all of the smaller balls have the same measurement; since the sum of these measurements is finite, the smaller balls must all have ${\displaystyle \mu }$ -measurement of zero.

See also!

• Cylinder set measure – way to generate a measure over product spacesPages displaying wikidata descriptions as a fallback
• Feldman–Hájek theorem – Theory in probability theory
• Gaussian measure#Infinite-dimensional spaces – Type of Borel measure
  • Structure theorem for Gaussian measures
• Projection-valued measure – Mathematical operator-value measure of interest in quantum mechanics and functional analysis
• Set function – Function from sets to numbers

References

• 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) (See section 1: Introduction)
• Oxtoby, John C.; Prasad, Vidhu S. (1978). "Homeomorphic measures on the Hilbert cube". Pacific Journal of Mathematics. 77 (2): 483–497. doi:10.2140/pjm.1978.77.483.