A remark on Bourgain’s distributional inequality on the Fourier spectrum of Boolean functions

Bourgain’s theorem says that under certain conditions a function \( f : \{0,1\}^n \to \{0,1\} \) can be approximated by a function \( g \) which depends only on a small number of variables. By following his proof we obtain a generalization for the case that there is a nonuniform product measure on the domain of \( f \).