Lebesgue-Stieltjes integral and young's inequality

Abstract

For non-decreasing real functions f and g; we consider the functional T(f,g; I,J) = ?I f(x) dg(x) + ?J g(x) df(x); where I and J are intervals with J ? I. In particular case with I = [a, t]; J = [a, s], a < s ? t and g(x) = x; this reduces to the expression in classical Young's inequality. We survey some properties of Lebesgue-Stieltjes integrals and present a simple proof for change of variables. Further, we formulate a version of Young's inequality with respect to arbitrary positive measure on real line and discuss applications in probability and number theory.