By Subhash Sharma

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**Example text**

If G acts properly on l( there exists a modulator plier. A subgroup K of implies that gKg- 1 type G/K with K m having l( (k) l( l( k € K. as its associated multi- o regular we say that (G,~) has regular orbit type. has regular orbit type, where the be any multiplier fulfilling 1, then to every multiplier G is said to be regular, if 9K9- 1 ~ K, g € G, K. 3. Suppose that (G,~) orbits are homeomorphic to G/K. Let ~, 31 Then there exists a modulator with as its associated multio ~ plier. The importance of the concept of modulator lies in the fact that by means of modulators it is possible to construct any relatively invariant measure - in particular, an invariant measure - on other relatively invariant measure on measure ~l( l(.

Can be considered as a sphere in IR is also clear by noting that ~ is the intersection between the sphere {x € IRnl

4, ~ = (x € ~nls(x) be the restriction to ant with multiplier It is clear that IAI. Furthermore, easily shown to be a modulator with l( is positive definite} A is relatively invari- m(x) = Is(x) 11/2 = Is(x) 1- 1 / 2 dA(x) is an invariant measure on ~. is as associated multiplier. It follows that ~(x) an. 2. CCn,Hl-invariant measure. A = l«(a,~» be ~n. as associated multiplier. Hence is an invariant measure on let A of Lebesgue measure on o 32 By the above mentioned existence and uniqueness result for invariant measures there exist measures a and ~ on G which are invariant, respectively, under left action 0 and right action c of G on itself, and these measures are termed left invariant and right invariant, respectively.