Minimal measure was introduced to study the dynamics of positive definite Hamiltonian in the last nineties, in terms of first homology or cohomology class. For autonomous system, they are supported on the level set not below the Ma\~n\'e critical value. When the fundamental group of the configuration manifold is larger than the first homologous group, it makes difference if we define minimal measure in terms of elements of the fundamental group. Indeed, more minimal measures are found, which is overlooked in Mather theory. In particular, minimal measure for commutator is discovered, it exists not only on the level set above the critical value, but also below the value. We also give a simple proof about the hyperbolicity of periodic orbit if it supports some minimal measure.