This is joint work with Y. Canzani, B. Clarke, N. Kamran, L. Silberman and J. Taylor. We define Gaussian measures on manifolds of metrics with the fixed volume form. We next compute the moment generating function for the L2 (Ebin) distance to the reference metric. We also prove integrability results (with respect to our measures) for functionals such as Laplace eigenvalues, diameter, and volume entropy.