In this paper, we consider a coherent theory about the asymptotic representations for a family of inequality indices called Theil-Like Inequality Measures ( TLIM ), within a Gaussian field. The theory uses the functional empirical process approach. We provide the finite-distribution and uniform asymptotic normality of the element of the TLIM class in a unified approach rather than in a case by case one. The results are then applied to some UEMOA countries databases.