Infinite-dimensional Lebesgue measure

An infinite-dimensional Lebesgue measure (or Lebesgue-like measure) is a measure defined on an infinite-dimensional Banach space, which shares certain properties with the Lebesgue measure defined on finite-dimensional spaces.

The usual Lebesgue measure does not generalize to all infinite-dimensional spaces, because any translation-invariant Borel measure on an infinite-dimensional separable Banach space is either infinite on all sets or zero on all sets. However, there are examples of Lebesgue-like measures when either the space is not separable (such as the Hilbert cube), or when one of the characteristic properties of the Lebesgue measure is relaxed.

Motivation

The Lebesgue measure ${\displaystyle \lambda }$ on the Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ is locally finite, strictly positive, and translation-invariant. That is:

• every point ${\displaystyle x}$ in ${\displaystyle \mathbb {R} ^{n}}$ has an open neighborhood ${\displaystyle N_{x}}$ with finite measure: ${\displaystyle \lambda (N_{x})<+\infty ;}$
• every non-empty open subset ${\displaystyle U}$ of ${\displaystyle \mathbb {R} ^{n}}$ has positive measure: ${\displaystyle \lambda (U)>0;}$ and
• if ${\displaystyle A}$ is any Lebesgue-measurable subset of ${\displaystyle \mathbb {R} ^{n},}$ and ${\displaystyle h}$ is a vector in ${\displaystyle \mathbb {R} ^{n},}$ then all translates of ${\displaystyle A}$ have the same measure: ${\displaystyle \lambda (A+h)=\lambda (A).}$

Motivated by their geometrical significance, constructing measures satisfying the above set properties, for infinite-dimensional spaces such as the ${\displaystyle L^{p}}$ spaces or path spaces is still an open and active area of research.

Non-Existence Theorem in Separable Banach spaces

Let ${\displaystyle X}$ 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. Equivalently, there is no locally finite, strictly positive, translation invariant measure on ${\displaystyle X}$.^[1]

Proof

Let ${\displaystyle X}$ be an infinite-dimensional, separable Banach space equipped with a locally finite translation-invariant measurement ${\displaystyle \mu .}$ To prove that ${\displaystyle \mu }$ is the trivial measure, it is sufficient and necessary to show that ${\displaystyle \mu (X)=0.}$

Like every separable metric space, ${\displaystyle X}$ is a Lindelöf space, which means that every open cover of ${\displaystyle X}$ has a countable subcover. It is, therefore, enough to show that there exists some open cover of ${\displaystyle X}$ by null sets because by choosing a countable subcover, the σ-subadditivity of ${\displaystyle \mu }$ implies that ${\displaystyle \mu (X)=0.}$

Using local finiteness, suppose that for some ${\displaystyle r>0,}$ the open ball ${\displaystyle B(r)}$ of radius ${\displaystyle r}$ has a finite ${\displaystyle \mu }$-measure. Since ${\displaystyle X}$ is infinite-dimensional, by Riesz's lemma there is an infinite sequence of pairwise disjoint open balls ${\displaystyle B_{n}(r/4),}$ ${\displaystyle n\in \mathbb {N} }$, of radius ${\displaystyle r/4,}$ with all the smaller balls ${\displaystyle B_{n}(r/4)}$ contained within ${\displaystyle B(r).}$ By translation invariance, all the smaller balls have the same measure, and since the sum of these measurements is finite, the smaller balls must all have ${\displaystyle \mu }$-measure zero.

Since ${\displaystyle r}$ was arbitrary, every open ball in ${\displaystyle X}$ has zero measure, and taking a cover of ${\displaystyle X}$ which is the set of all open balls completes the proof.

Nontrivial measures

The following are examples where a notion of an infinite-dimensional Lebesgue measure exists, once the conditions of the above theorem are loosened.

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.^[2]

The Hilbert cube carries the product Lebesgue measure^[3] 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.^{[citation needed]}

See also

• Cylinder set measure – way to generate a measure over product spacesPages displaying wikidata descriptions as a fallback
• Cameron–Martin theorem – Theorem defining translation of Gaussian measures (Wiener measures) on Hilbert spaces.
• Feldman–Hájek theorem – Theory in probability theory
• Gaussian measure#Infinite-dimensional spaces – Type of Borel measure
  • Structure theorem for Gaussian measures – Mathematical theorem
• Projection-valued measure – Mathematical operator-value measure of interest in quantum mechanics and functional analysis
• Set function – Function from sets to numbers

References