infinite-dimensional Lebesgue measure

The Lebesgue measure $\backslash lambda$ on the Euclidean space $\backslash Reals^n$ is locally finite, strictly positive, and translation-invariant. That is:

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

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.

Let $X$ be an infinite-dimensional, separable Banach space. Then, the only locally finite and translation invariant Borel measure $\backslash mu$ on $X$ is a trivial measure. Equivalently, there is no locally finite, strictly positive, and translation invariant measure on $X$.{{cite journal |last=Oxtoby |first=John C. |date=1946 |title=Invariant measures in groups which are not locally compact |journal=Trans. Amer. Math. Soc. |volume=60 |pages=216 |doi=10.1090/S0002-9947-1946-0018188-5}}

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) a measure that perfectly matches the properties of a finite dimensional Lebesgue measure does not exist.

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

Like every separable metric space, $X$ is a Lindelöf space, 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 $\backslash mu$ will imply that $\backslash mu(X)\; =\; 0.$

Using local finiteness of the measure $\backslash mu$, suppose that for some $r\; >\; 0,$ the open ball $B(r)$ of radius $r$ has a finite $\backslash mu$-measure. Since $X$ is infinite-dimensional, by Riesz's lemma there is an infinite sequence of pairwise disjoint open balls $B\_n(r/4),$ $n\; \backslash in\; \backslash N$, 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 $\backslash mu$-measure, and since the infinite sum of these finite $\backslash mu$-measures are finite, the cover's balls must all have $\backslash mu$-measure zero.

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

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 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.{{cite journal|author=Hunt, Brian R. and Sauer, Tim and Yorke, James A.|title=Prevalence: a translation-invariant "almost every" on infinite-dimensional spaces|journal=Bull. Amer. Math. Soc. (N.S.)|volume=27|year=1992|pages=217–238|doi=10.1090/S0273-0979-1992-00328-2|issue=2|arxiv=math/9210220|bibcode=1992math.....10220H|s2cid=17534021}}

The Hilbert cube carries the product Lebesgue measure{{cite journal |last1=Oxtoby |first1=John C. |last2=Prasad |first2=Vidhu S. |title=Homeomorphic Measures on the Hilbert Cube |journal=Pacific J. Math. |date=1978 |volume=77 |issue=2 |pages=483–497 |doi=10.2140/pjm.1978.77.483 |url=https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-77/issue-2/Homeomorphic-measures-in-the-Hilbert-cube/pjm/1102806462.full}} and the compact topological group given by the Tychonoff product of an infinite number of copies of the circle group 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|date=January 2024}}

• {{annotated link|Cylinder set measure}}
• {{annotated link|Cameron–Martin theorem}}
• {{annotated link|Feldman–Hájek theorem}}
• {{annotated link|Gaussian measure#Infinite-dimensional spaces}}
• {{annotated link|Structure theorem for Gaussian measures}}
• {{annotated link|Projection-valued measure}}
• {{annotated link|Set function}}

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