Various expressions for modulus of random convexity

We first prove various kinds of expressions for modulus of random convexity by using an L°(F,.)-valued function.s intermediate value theorem and the well known Hahn-Banach theorem for almost surely bounded random linear functionals, then establish some basic properties including continuity for modulus of random convexity. In particular, we express the modulus of random convexity of a special random normed module L°(F,X) derived from a normed space X by the classical modulus of convexity of X.