SMOOTH POINT MEASURES AND DIFFEOMORPHISM GROUPS

Let X=R~d(d>1).Consider the unitary representations of Diff(X)given by quasi-invariant measures under the action of Diff(X).The author proposes smooth point measuresas generalization of Poisson point measures and proves that every smooth point measure isquasi-invariant under the action of Diff(X)and if {U_g~i},i=1,2,are the unitary represen-tations of Diff(X)given by the smooth point measures μ_i,i=1,2,respectively,then {U_g~1}is unitarily equivalent to {U_g~2} iff μ_1 is equivalent to μ_2 as measure.

SMOOTH POINT MEASURES AND DIFFEOMORPHISM GROUPS

Let $[X = {R^d}(d > 1)]$. Consider the unitary representations of Diff(X) given by quasi-invariant measures under the action of Diff(X). The author proposes smooth point measures as generalization of Poisson point measures and proves that every smooth point measure is quasi-invariant under the action of Diff(X) and if $[{ U_g^i} ]$ ,i=1,2, are the unitar representations of Diff(X) given by the smooth point measures $[{mu _i}]$, i=l,2, respectively, then $[{ U_g^1} ]$ is unitarily equivalent to $[{ U_g^2} ]$ iff $[{mu _1}]$ is equivalent to $[{mu _2}]$ as measure.