An infinite-dimensional Lebesgue measure is a type of measure defined on an infinite-dimensional normed vector space, especially a Banach space, that shares properties with the Lebesgue measure on finite-dimensional spaces.
The usual Lebesgue measure cannot be simply 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. Nonetheless, there are meaningful instances of infinite-dimensional Lebesgue-like measures that occur when the infinite-dimensional space is not separable, as for the Hilbert cube, or when one of the defining properties of a finite-dimensional Lebesgue measure is relaxed.
The Lebesgue measure
λ
\Realsn
x
\Realsn
Nx
λ(Nx)<+infty;
U
\Realsn
λ(U)>0;
A
\Realsn,
h
\Realsn,
A
λ(A+h)=λ(A).
Lp
Let
X
\mu
X
X
G
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
\mu
\mu
\mu(X)=0.
Like every separable metric space,
X
X
X
\mu
\mu(X)=0.
Using local finiteness of the measure
\mu
r>0,
B(r)
r
\mu
X
Bn(r/4),
n\in\N
r/4,
Bn(r/4)
B(r).
\mu
\mu
\mu
Since
r
X
\mu
X
\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.[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.