Let G be a connected reductive linear algebraic group defined over a (nondiscrete) locally compact field k of characteristic zero, and G be its group of k-rational points. If O(x) = {yxy': y e G} is the orbit of x, then it is known2 that O(x) is locally compact (in the Hausdorff topology of G) and homeomorphic to G/G,. The isotropy subgroup Gx is known to be unimodular ([2], p. 235) and so the space G/GX carries a G-invariant Radon measure dy*(y* =yGx). A question that seems to be of some importance in harmonic analysis is, whether this measure, transplanted to the orbit O(x), is a Radon measure in G, i.e., is finite for compact subsets of G. If this is the case then the integral