In functional analysis and measure theory, one may seek to extend the Lebesgue measure on ${\displaystyle \mathbb {R} ^{n}}$ to an analogous measure on an infinite dimensional separable Banach space. Without departing substantially from the essential properties of the Lebesgue measure, this is impossible. Indeed, any translation invariant Borel measure on an infinite dimensional separable Banach space is either infinite on all open sets, or the zero measure.

On the other hand, numerous examples of such measures exist once one allows non-separable Banach spaces, or allows the measure to not be translation invariant. In the same vein, one may extend the Lebesgue measure to various subsets of infinite-dimensional space, like the Hilbert cube.

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.

Separable Banach Spaces

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 ${\displaystyle X.}$

Nontrivial Measures

Here we describe examples where a notion of an infinite-dimensional Lebesgue measure exists, once one loosens the conditions of the above theorem.

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.^[1]

The Hilbert cube carries the product Lebesgue measure^[2] 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.

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