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In mathematics, aninfinite-dimensional Lebesgue measure is ameasure defined on infinite-dimensionalnormed vector spaces, such asBanach spaces, which resembles theLebesgue measure 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-invariantBorel measure on an infinite-dimensionalseparable 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 locally compact spaces like theHilbert cube, which is compact, or scenarios where some typical properties of finite-dimensional Lebesgue measures are modified or omitted.

The Lebesgue measure${\displaystyle \lambda }$ on theEuclidean space${\displaystyle {\mathbb{R}}^n}$ islocally finite,strictly positive, andtranslation-invariant. That is:

• every point${\displaystyle x}$ in${\displaystyle {\mathbb{R}}^n}$ has an open neighborhood${\displaystyle N_x}$ with finite measure:
• 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 orpath spaces is still an open and active area of research.

Let${\displaystyle X}$ be an infinite-dimensional,separable Banach space. Then, the only locally finite and translation invariantBorel measure${\displaystyle \mu }$ on${\displaystyle X}$ is atrivial measure. Equivalently, there is no locally finite, strictly positive, and translation invariant measure on${\displaystyle X}$.^[1]

More generally: on a non locally compactPolish group${\displaystyle G}$, there cannot exist aσ-finite andleft-invariant Borel measure.^[1]

This theorem implies that on an infinite dimensional separable Banach space (which cannot belocally compact) a measure that perfectly matches the properties of a finite dimensional Lebesgue measure does not exist.

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

Like every separablemetric space,${\displaystyle X}$ is aLindelöf 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, theσ-subadditivity of${\displaystyle \mu }$ will imply that${\displaystyle \mu (X)=0.}$

Using local finiteness of the measure${\displaystyle \mu }$, suppose that for some${\displaystyle r>0,}$ theopen ball${\displaystyle B(r)}$ of radius${\displaystyle r}$ has a finite${\displaystyle \mu }$-measure. Since${\displaystyle X}$ is infinite-dimensional, byRiesz's lemma there is an infinite sequence ofpairwise 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 cover's balls have the same${\displaystyle \mu }$-measure, and since the infinite sum of these finite${\displaystyle \mu }$-measures are finite, the cover's balls must all have${\displaystyle \mu }$-measure zero.

Since${\displaystyle r}$ was arbitrary, every open ball in${\displaystyle X}$ has zero${\displaystyle \mu }$-measure, and taking a cover of${\displaystyle X}$ which is the set of all open balls that completes the proof that${\displaystyle \mu (X)=0}$.

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 theabstract 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 atprevalent and shy sets.^[2]

TheHilbert cube carries theproduct Lebesgue measure^[3] and the compacttopological group given by theTychonoff product of an infinite number of copies of thecircle group is infinite-dimensional and carries aHaar 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]}

• Cylinder set measure
• Cameron–Martin theorem – Theorem describing translation of Gaussian 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 – Measure used in functional analysis
• Set function – Function from sets to numbers

• Banach spaces
• Measure theory
• Theorems in measure theory
• Theorems in mathematical analysis

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