Abstract: As a generalization of Guassian measures, CM-measures on R^∞ are presented here. We study the cores of Laplace-Beltrami operators with respeot to them and the commutative relations between differential operators on R^∞ and the symmetric diffusion semi-groups generated by the Lallace-Beltrami operators. Using these results, the Littlwood-Paley inequlities will be established in the following paper on more general measure spaces than Wiener space.