Does There Exist a~Lebesgue Measure in the Infinite-Dimensional Space?

A. M. Vershik

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analysis and singularities. Part 2, Tome 259 (2007), pp. 256-281

Résumé

We consider sigma-finite measures in the space of vector-valued distributions on a manifold $X$ with the characteristic functional $\Psi(f)=\exp\bigl\{-\theta\int_X\ln\lVert f(x)\rVert\,dx\bigr\}$, $\theta>0$. The collection of such measures constitutes a one-parameter semigroup relative to $\theta$. In the case of scalar distributions and $\theta=1$, this measure may be called the infinite-dimensional Lebesgue measure. We prove that the weak limit of the Haar measures on the Cartan subgroups of the groups $\operatorname{SL}(n,\mathbb R)$, when $n$ tends to infinity, is that infinite-dimensional Lebesgue measure. This measure is invariant under the linear action of some infinite-dimensional abelian group that can be viewed as an analog of an infinite-dimensional Cartan subgroup; this fact can serve as a justification of the name Lebesgue as a valid name for the measure in question. Application to the representation theory of current groups was one of the reasons to define this measure. The measure is also closely related to the Poisson–Dirichlet measures well known in combinatorics and probability theory. The only known example of analogous asymptotic behavior of the uniform measure on the homogeneous manifold is the classical Maxwell–Poincaré lemma, which states that the weak limit of uniform measures on the Euclidean spheres of appropriate radius, as dimension tends to infinity, is the standard infinite-dimensional Gaussian measure. Our situation is similar, but all the measures are no more finite but sigma-finite. The result raises an important question about the existence of other types of interesting asymptotic behavior of invariant measures on the homogeneous spaces of Lie groups.