Infinite-dimensional analogue of the Lebesgue measure and some properties of the gamma process

The talk presents the results of the joint work with A. Vershik and M. Yor. We define a sigma-finite measure on the space of discrete measures on a measurable space which is equivalent to the law of the classical gamma process and is invariant under an infinite-dimensional group of multiplicators. This measure was first discovered in the works by Gelfand–Graev–Vershik on the representation theory of current groups, but we construct it explicitly using some properties of the gamma process. The above invariance property is a natural generalization of the corresponding property of the Lebesgue measure in $\mathbf R^n$, and this allows us to call the constructed measure the infinite-dimensional Lebesgue measure. It enjoys many distinguished properties, some of them will be considered in the talk.