Infinite-dimensional Lebesgue measure

In mathematics, it is a theorem that there is no analogue of Lebesgue measure on an infinite-dimensional Banach space. Other kinds of measures are therefore used on infinite-dimensional spaces: often, the abstract Wiener space construction is used. Alternatively, one may consider Lebesgue measure on finite-dimensional subspaces of the larger space and consider so-called prevalent and shy sets.

Compact sets in Banach spaces may also carry natural measures: the Hilbert cube, for instance, carries the product Lebesgue measure. In a similar spirit, 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.

Motivation

It can be shown that Lebesgue measure λⁿ on Euclidean space Rⁿ is locally finite, strictly positive and translation-invariant, explicitly:

• every point x in Rⁿ has an open neighbourhood N_x with finite measure λⁿ(N_x) < +∞;
• every non-empty open subset U of Rⁿ has positive measure λⁿ(U) > 0; and
• if A is any Lebesgue-measurable subset of Rⁿ, T_h : Rⁿ → Rⁿ, T_h(x) = x + h, denotes the translation map, and (T_h)_∗(λⁿ) denotes the push forward, then (T_h)_∗(λⁿ)(A) = λⁿ(A).

Geometrically speaking, these three properties make Lebesgue measure very nice to work with. 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 nice to have a similarly nice measure to work with. Unfortunately, this is not possible.

Statement of the theorem

Let (X, ||·||) be an infinite-dimensional, separable Banach space. Then the only locally finite and translation-invariant Borel measure μ on X is the trivial measure, with μ(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 μ. Using local finiteness, suppose that, for some δ > 0, the open ball B(δ) of radius δ has finite μ-measure. Since X is infinite-dimensional, there is an infinite sequence of pairwise disjoint open balls B_n(δ/4), n ∈ N, of radius δ/4, with all the smaller balls B_n(δ/4) contained within the larger ball B(δ). 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 μ-measure zero. Now, since X is separable, it can be covered by a countable collection of balls of radius δ/4; since each such ball has μ-measure zero, so must the whole space X, and so μ is the trivial measure.

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. doi:10.1090/S0273-0979-1992-00328-2.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).