A Study and Comparison of the Stieltjes and Lebesgue-Stieltjes Integration Theories.

A generalization of the Riemann integral commonly referred to as the Stieltjes integral is studied. The three definitions of this integral found in the literature are compared and found to be dissimilar. The most general of them, the Generalized Stieltjes integral, is chosen for further study. Essential properties of this integral and criteria for establishing Stieltjes integrability that is, whether the integral of a function over an interval assumes a finite value are examined. Next, the Lebesgue-Stieltjes integral, a generalization of the Lebesgue integral, is studied. Essential properties of this integral are determined and criteria for establishing Lebesgue-Stieltjes integrability are established. Stieltjes and Lebesgue-Stieltjes integration are compared. Modified author abstract