Two-scale convergence with respect to measures and homogenization of monotone operators

In 1989 Nguetseng introduced two-scale convergence, which now is a frequently used tool in homogenization of partial differential operators. In this paper we discuss the notion of two-scale convergence with respect to measures. We make an exposition of the basic facts of this theory and develope it in various ways. In particular, we consider both variable Lp spaces and variable Sobolev spaces. Moreover, we apply the results to a homogenization problem connected to a class of monotone operators.