In mathematics, an infinite-dimensional Lebesgue measure is a measure defined on infinite-dimensional normed vector spaces, such as Banach spaces, which resembles the Lebesgue 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-invariant Borel measure on an infinite-dimensional separable 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 non-separable spaces like the Hilbert cube, or scenarios where some typical properties of finite-dimensional Lebesgue measures are modified or omitted.
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
be an infinite-dimensional, separable Banach space.
Then, the only locally finite and translation invariant Borel measure
Equivalently, there is no locally finite, strictly positive, and translation invariant measure on
[1] More generally: on a non locally compact Polish group
, there cannot exist a σ-finite and left-invariant Borel measure.
[1] 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.
be an infinite-dimensional, separable Banach space equipped with a locally finite translation-invariant measure
is the trivial measure, it is sufficient and necessary to show that
is a Lindelöf space, which means that every open cover of
It is, therefore, enough to show that there exists some open cover of
by null sets because by choosing a countable subcover, the σ-subadditivity of
is infinite-dimensional, by Riesz's lemma there is an infinite sequence of pairwise disjoint open balls
By translation invariance, all the cover's balls have the same
which is the set of all open balls that completes the proof that
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.
[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 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.