Infinite-dimensional Lebesgue measure

An infinite-dimensional Lebesgue measure is a type of measure defined on an infinite-dimensional normed vector space, more specifically a Banach space. It shares properties with the Lebesgue measure on finite-dimensional spaces.

However, the usual Lebesgue measure cannot be extended to all infinite-dimensional spaces. This limitation arises because any translation-invariant Borel measure on an infinite-dimensional separable Banach space is always either infinite for all sets or zero for all sets. Despite this, there are some instances of Lebesgue-like measures. These occur when the space is not separable, such as the Hilbert cube, or when one of the key properties of the Lebesgue measure is relaxed.

Motivation
The Lebesgue measure $$\lambda$$ on the Euclidean space $$\Reals^n$$ is locally finite, strictly positive, and translation-invariant. That is: 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.
 * 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: $$\lambda(A+h) = \lambda(A).$$

Statement of the theorem
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 can not be locally compact - a Lebesgue measure does not exist.

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 $$\mu$$ on $$X$$ is a trivial measure. Equivalently, there is no locally finite, strictly positive, and translation invariant measure on $$X$$.

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

Like every separable metric space, $$X$$ 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 $$\mu$$ implies that $$\mu(X) = 0.$$

Using local finiteness, suppose that for some $$r > 0,$$ the open ball $$B(r)$$ 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 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 smaller balls have the same measure, and since the sum of these measurements is finite, the smaller balls must all have $$\mu$$-measure zero.

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

Nontrivial measures
Here are some examples of infinite-dimensional Lebesgue measures that can exist if the conditions of the above theorem are relaxed.

There are other measures that support entirely separable Banach spaces. One example is the abstract Wiener space construction, similar to products of Gaussian measures. Another approach is to consider a Lebesgue measurement of finite-dimensional subspaces within the larger space and look at prevalent and shy sets.

The Hilbert cube carries the product Lebesgue measure 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.