CHAPTER XIII - THE STIELTJES INTEGRAL

Abstract

This chapter discusses the Stieltjes integral. The generalization of the Eiemann integral is the Stieltjes integral. The Stieltjes integral is a generalization of the Eiemann integral and reduces to it for the particular case when g(x) = x. If the number of jumps is denumerable, the Stieltjes integral goes over into an infinite series. In all cases, the Stieltjes integral introduces nothing essentially new and reduces to already-known ideas. In general, however, this is not the case and it can be stated that if g(x) is a function of finite variation continuous but not absolutely continuous, then the Stieltjes integral cannot be reduced in a way as before to either a Lebesgue integral or to a sum or infinite series.