Elsevier

Journal of Functional Analysis

Volume 185, Issue 1, 10 September 2001, Pages 274-296
Journal of Functional Analysis

Regular Article
An Infinite-Dimensional Analogue of the Lebesgue Measure and Distinguished Properties of the Gamma Process

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Abstract

We define a one-parameter family Lθ of sigma-finite (finite on compact sets) measures in the space of distributions. These measures are equivalent to the laws of the classical gamma processes and invariant under an infinite-dimensional abelian group of certain positive multiplicators. This family of measures was first discovered by Gelfand–Graev–Vershik in the context of the representation theory of current groups; here we describe it in direct terms using some remarkable properties of the gamma processes. We show that the class of multiplicative measures coincides with the class of zero-stable measures which is introduced in the paper. We give also a new construction of the canonical representation of the current group SL(2, R)X.

Keywords

infinite-dimensional Lebesgue measure
gamma process
sigma-finite invariant zero-stable measures

Cited by (0)

f1

E-mail: [email protected]

f2

E-mail: [email protected]

1

Partially supported by RFBR Grant 00-15-96060, DFG-RFBR Grant 99-01-04027, and INTAS Grant 99-1317.

2

Partially supported by RFBR Grant 99-01-00098 and an NWO grant.