无限维空间中是否存在勒贝格测度？
Does there exist the Lebesgue measure in the infinite-dimensional space?
摘要：
利用Laplace变换$$Ψ(f)=\exp\{-θ\int_X\ln||f(x)||dx\}, θ>0.$$研究了流形$X$上向量值分布空间中的σ-有限测度，并讨论了当$n$趋于无穷大时，群$SL(n,{\Bbb R})$的Cartan子群上Haar测度的弱极限.极限中的测度称为{\it无穷维勒贝格测度}。它在类似于Cartan子群的无穷维Abel群的线性作用下是不变的。该措施也是密切相关的泊松-狄利克雷措施众所周知的组合和概率论。在齐次流形上一致测度的类似渐近行为的唯一已知例子是{\it经典麦克斯韦-庞加莱引理}，它断言当维数趋于无穷大时，在适当半径的欧氏球面上一致测度的弱极限是标准的无限维高斯测度和白噪声，但在我们的情况下，所有的测度不再是有限的，而是σ-有限的。
英文摘要：
We study the sigma-finite measures in the space of vector-valued distributions on the manifold $X$ with Laplace transform $$Ψ(f)=\exp\{-θ\int_X\ln||f(x)||dx\}, θ>0.$$ We also consider the weak limit of Haar measures on the Cartan subgroup of the group $SL(n,{\Bbb R})$ when $n$ tends to infinity. The measure in the limit is called {\it infinite dimensional Lebesgue measure}. It is invariant under the linear action of some infinite-dimensional Abelian group which is an analog of Cartan subgroup. The measure also is closely related to the Poisson--Dirichlet measures well known in combinatorics and probability theory. The only known example of the analogous asymptotical behavior of the uniform measure on the homogeneous manifold is {\it classical Maxwell-Poincaré lemma} which asserts that the weak limit of uniform measures on the Euclidean sphere of appropriate radius as dimension tends to infinity is the standard infinite-dimensional Gaussian measure and white noise, but in our situation all the measures are no more finite but sigma-finite.
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