  In this paper we construct short time classical solutions to a class of
master equations in the presence of non-degenerate individual noise arising in
the theory of mean field games. The considered Hamiltonians are non-separable
and $local$ functions of the measure variable, therefore the equation is
restricted to absolutely continuous measures whose densities lie in suitable
Sobolev spaces. Our results hold for smooth enough Hamiltonians, without any
additional structural conditions as convexity or monotonicity.
