There is no infinite-dimensional Lebesgue measure
Encyclopedia
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.
Rn is locally finite
, strictly positive
and translation
-invariant
, explicitly:
Geometrically
speaking, these three properties make Lebesgue measure very nice to work with. When we consider an infinite-dimensional space such as an Lp 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.
, 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.
and a Lindelöf space
; hence, 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, it is a theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
that there is no analogue of Lebesgue measure
Lebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...
on an infinite-dimensional Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
. Other kinds of measures are therefore used on infinite-dimensional spaces: often, the abstract Wiener space
Abstract Wiener space
An abstract Wiener space is a mathematical object in measure theory, used to construct a "decent" measure on an infinite-dimensional vector space. It is named after the American mathematician Norbert Wiener...
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
Prevalent and shy sets
In mathematics, the notions of prevalence and shyness are notions of "almost everywhere" and "measure zero" that are well-suited to the study of infinite-dimensional spaces and make use of the translation-invariant Lebesgue measure on finite-dimensional real spaces...
.
Compact sets in Banach spaces may also carry natural measures: the Hilbert cube
Hilbert cube
In mathematics, the Hilbert cube, named after David Hilbert, is a topological space that provides an instructive example of some ideas in topology...
, for instance, carries the product Lebesgue measure
Product measure
In mathematics, given two measurable spaces and measures on them, one can obtain the product measurable space and the product measure on that space...
. In a similar spirit, the compact topological group
Topological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
given by the Tychonoff product of infinitely many copies of the circle group is infinite-dimensional, and carries a Haar measure
Haar measure
In mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of locally compact topological groups and subsequently define an integral for functions on those groups....
that is translation-invariant.
Motivation
It can be shown that Lebesgue measure λn on Euclidean spaceEuclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
Rn is locally finite
Locally finite measure
In mathematics, a locally finite measure is a measure for which every point of the measure space has a neighbourhood of finite measure.-Definition:...
, strictly positive
Strictly positive measure
In mathematics, strict positivity is a concept in measure theory. Intuitively, a strictly positive measure is one that is "nowhere zero", or that it is zero "only on points".-Definition:...
and translation
Translation (geometry)
In Euclidean geometry, a translation moves every point a constant distance in a specified direction. A translation can be described as a rigid motion, other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant vector to every point, or...
-invariant
Invariant measure
In mathematics, an invariant measure is a measure that is preserved by some function. Ergodic theory is the study of invariant measures in dynamical systems...
, explicitly:
- every point x in Rn has an openOpen setThe concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
neighbourhoodNeighbourhood (mathematics)In topology and related areas of mathematics, a neighbourhood is one of the basic concepts in a topological space. Intuitively speaking, a neighbourhood of a point is a set containing the point where you can move that point some amount without leaving the set.This concept is closely related to the...
Nx with finite measure λn(Nx) < +∞; - every non-empty open subset U of Rn has positive measure λn(U) > 0; and
- if A is any Lebesgue-measurable subset of Rn, Th : Rn → Rn, Th(x) = x + h, denotes the translation map, and (Th)∗(λn) denotes the push forwardPushforward measureIn measure theory, a pushforward measure is obtained by transferring a measure from one measurable space to another using a measurable function.-Definition:...
, then (Th)∗(λn)(A) = λn(A).
Geometrically
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
speaking, these three properties make Lebesgue measure very nice to work with. When we consider an infinite-dimensional space such as an Lp space
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
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 measureTrivial measure
In mathematics, specifically in measure theory, the trivial measure on any measurable space is the measure μ which assigns zero measure to every measurable set: μ = 0 for all A in Σ.-Properties of 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 Bn(δ/4), n ∈ N, of radius δ/4, with all the smaller balls Bn(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 a separable normed space, it is also a second-countable spaceSecond-countable space
In topology, a second-countable space, also called a completely separable space, is a topological space satisfying the second axiom of countability. A space is said to be second-countable if its topology has a countable base...
and a Lindelöf space
Lindelöf space
In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of compactness, which requires the existence of a finite subcover....
; hence, 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.