Before revision:  We show that such probability measure exists if and only if the \alpha average of  so-called G-expectation of bounded uniformly continuous and bounded functions with respect to the measures in \Gamma is less than their \beta average.
After revision:  We show that such probability measure exists if and only if the \alpha average of the so-called G-expectation of bounded uniformly continuous and bounded functions with respect to the measures in \Gamma is less than their \beta average.