Stochastic metastability is stability weakened by chance. In a definition of stability certain conditions must be satisfied for sufficiently small displacements. Corresponding to a definition of stability we introduce a definition of (stochastic) metastability, in which the same conditions must be satisfied with probability approaching unity as the maximum displacement approaches zero. Measurable sets are essential to metastability as open sets are to stability. Because of Arnol'd diffusion, the invariant tori of conservative Hamiltonian systems of $n\geq $ 3 degrees of freedom are not usually dynamically stable with respect to changes in initial conditions. However, we prove using the Lebesgue density theorem that if the proper invariant tori of phase space fill a bounded domain of positive measure, then they are almost all dynamically metastable. Metastability is difficult to distinguish from stability in physical systems or numerical experiments, so the result is consistent with the difficulty of observing Arnol'd diffusion in the neighbourhood of an invariant torus in practice. Probability and measure are essential to the general theory of stability of nonlinear systems.