Infinite-dimensional Lebesgue measure

In functional analysis and measure theory, there is a folklore claim that there is no analogue of the Lebesgue measure on an infinite-dimensional Banach space. The claim states that there is no translation invariant measure on a separable Banach space—because if any ball has a nonzero non-infinite volume, a slightly smaller ball has zero volume, and countable many such smaller balls cover the space. The folklore statement, however, is entirely false. The countable product of Lebesgue measures is translation 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 with support entirely on 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 measure on finite-dimensional subspaces of 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 infinitely many copies of the circle group 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 $\lambda$ on Euclidean space $\mathbb {R} ^{n}$ is locally finite, strictly positive and translation-invariant, explicitly:

every point $x$ in $\mathbb {R} ^{n}$ has an open neighbourhood $N_{x}$ with finite measure ${\displaystyle \lambda (N_{x})<+\infty$
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 $\mathbb {R} ^{n},$ $T_{n}:\mathbb {R} ^{n}\to \mathbb {R} ^{n},$ $T_{h}(x)=x+h,$ denotes the translation map, and $(T_{h})_{*}(\lambda )$ denotes the push forward, then $(T_{h})_{*}(\lambda )(A)=\lambda (A).$

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

Statement of the theorem

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

Proof of the theorem

Let $X$ be an infinite-dimensional, separable Banach space equipped with a locally finite, translation-invariant measure $\mu .$

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

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

Using local finiteness, suppose that, for some $r>0,$ the open ball $B(r)$ of radius $r$ has 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 the larger ball $B(r).$ By translation-invariance, all of the smaller balls have the same measure; since the sum of these measures is finite, the smaller balls must all have $\mu$ -measure zero.

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.

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Infinite-dimensional Lebesgue measure

Motivation

Statement of the theorem

Proof of the theorem

See also

References