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In mathematics, an infinite-dimensional Lebesgue measure is a measure defined on infinite-dimensional
normed vector spaces The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, France, in 1898. The shipyard boomed in the period before World War I (1914–18), but struggled in the inter-war p ...
, such as
Banach spaces In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
, which resembles the
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
used in finite-dimensional spaces. However, the traditional Lebesgue measure cannot be straightforwardly extended to all infinite-dimensional spaces due to a key limitation: any translation-invariant
Borel measure In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. ...
on an infinite-dimensional separable Banach space must be either infinite for all sets or zero for all sets. Despite this, certain forms of infinite-dimensional Lebesgue-like measures can exist in specific contexts. These include non-separable spaces like the
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. Furthermore, many interesting topological spaces can be embedded in the Hilbert cube; that is, ca ...
, or scenarios where some typical properties of finite-dimensional Lebesgue measures are modified or omitted.


Motivation

The Lebesgue measure \lambda on the
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
\Reals^n is locally finite, strictly positive, and translation-invariant. That is: * every point x in \Reals^n has an open neighborhood N_x with finite measure: \lambda(N_x) < + \infty; * every non-empty open subset U of \Reals^n has positive measure: \lambda(U) > 0; and * if A is any Lebesgue-measurable subset of \Reals^n, and h is a vector in \Reals^n, then all translates of A have the same measure: \lambda(A+h) = \lambda(A). Motivated by their geometrical significance, constructing measures satisfying the above set properties for infinite-dimensional spaces such as the L^p spaces or path spaces is still an open and active area of research.


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 In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. ...
\mu on X is a
trivial measure In mathematics, specifically in measure theory, the trivial measure on any measurable space (''X'', Σ) is the measure ''μ'' which assigns zero measure to every measurable set: ''μ''(''A'') = 0 for all ''A'' in Σ. Properties of the trivial mea ...
. Equivalently, there is no locally finite, strictly positive, and translation invariant measure on X.


Statemant for non locally compact Polish groups

More generally: on a non locally compact Polish group G, there cannot exist a σ-finite and left-invariant Borel measure. This theorem implies that on an infinite dimensional separable Banach space (which cannot be
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which e ...
) a measure that perfectly matches the properties of a finite dimensional Lebesgue measure does not exist.


Proof

Let X be an infinite-dimensional, separable Banach space equipped with a locally finite translation-invariant measure \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 In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
, X is a
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'' sub ...
, 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 will imply that \mu(X) = 0. Using local finiteness of the measure \mu, suppose that for some r > 0, the
open ball In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defin ...
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 In set theory in mathematics and Logic#Formal logic, formal logic, two Set (mathematics), sets are said to be disjoint sets if they have no element (mathematics), element in common. Equivalently, two disjoint sets are sets whose intersection (se ...
open balls B_n(r/4), n \in \N, of radius r/4, with all the smaller balls B_n(r/4) contained within B(r). By translation invariance, all the cover's balls have the same \mu-measure, and since the infinite sum of these finite \mu-measures are finite, the cover's balls must all have \mu-measure zero. Since r was arbitrary, every open ball in X has zero \mu-measure, and taking a cover of X which is the set of all open balls that completes the proof that \mu(X) = 0.


Nontrivial measures

Here are some examples of infinite-dimensional Lebesgue measures that can exist if the conditions of the above theorem are relaxed. One example for an entirely separable Banach space is the abstract Wiener space construction, similar to a product of Gaussian measures (which are not translation invariant). Another approach is to consider a Lebesgue measure of finite-dimensional subspaces within the larger space and look at prevalent and shy sets. The
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. Furthermore, many interesting topological spaces can be embedded in the Hilbert cube; that is, ca ...
carries the product Lebesgue measure and the compact
topological group In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures ...
given by the Tychonoff product of an infinite number of copies of the
circle group In mathematics, the circle group, denoted by \mathbb T or , is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers \mathbb T = \. The circle g ...
is infinite-dimensional and carries a
Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This Measure (mathematics), measure was introduced by Alfr� ...
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.


See also

* * * * ** * *


References

{{Functional analysis Articles containing proofs Banach spaces Measure theory Theorems in measure theory Theorems in mathematical analysis