Let F(t) be a probability distribution function, its Stieltjes transform is defined by S_{F}(z)=int_{-infty}^{infty}frac{1}{t-z}dF(t), where z=x+iyin$ {f C}, y>0. In this thesis, we are interested in what f being the Stieltjes transform of some F. That is, we want to know what conditions f has, then f(z) can be written by int_{-infty}^{infty}frac{1}{t-z}dF(t).

The purpose of the thesis is to investigate an approach to Lebesgue-Stieltjes measure and integration.

In this thesis, we study the Stieltjes transforms of the probability distribution functions and compare them with the characteristic functions of the probability distribution functions simultaneously. In section 1 and section 2, we introduce briefly the Stieltjes transforms. In section 3, we conclude that the Stieltjes transform is similar to the complexion of symmetry under the condition of symmetric probability distribution functions. In section 4, we discuss the relation between Stieltjes transforms of probability distribution functions and the density of probability distribution functions. We also show that the nth derivative of Stieltjes transform is uniformly continuous on the upper complex plane.

Thesis (M. Phil.)--Hong Kong University of Science and Technology, 2002. / Includes bibliographical references (leaves 40). Also available in electronic version. Access restricted to campus users.

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This paper is a study of the Stieltjes integral, a generalization of the Riemann integral normally studied in introductory calculus courses. The purpose of the paper is to investigate many of the basic manipulative properties of the integral.

It is the purpose of this paper to define a Lebesgue integral over a measurable set, the integration being performed with respect to a monotone non-decreasing function as in the Stieltjes integral, and to develop a few of the fundamental properties of such an integral.

Provided in this thesis is the definition of Riemann Stieltjes Integration and properties of this integral. The Riemann Stieltjes Integral is compared with the Riemann Integral. Also, applications and limitations of the Riemann Stieltjes Integral are given.

Since F. Riesz showed in 1909 that the dual of C[0, 1] is BV[0, 1] (the functions of bounded variation on [0, 1] with II g IIBV = V(g)) via the Stieltjes integral, obtaining representations for linear operators in various settings has been a problem of interest. This paper shows the historical manner of representations, the road map type theorems and representations obtained via the v-integral. (44 pages)

O intuito deste trabalho é fazer uma análise sobre o processo de integração de funções. Existem muitas generalizações do conceito de integração abordado inicialmente por meio da integral de Riemann, como por exemplo, a integral de Riemann-Stieltjes, Lebesgue, Henstock-Kurzweil entre outras. Abordaremos especialmente a integral de Riemann-Stieltjes, e mostraremos a limitação da integral de Riemann no estudo de convergência de funções, indicando a necessidade de se generalizar o processo de integração. Faremos uma aplicação da integral de Riemann-Stieltjes no estudo de variáveis aleatórias e apresentamos uma proposta de abordagem, para a sala de aula, sobre o deslocamento e distância percorrida por um objeto em movimento retilíneo uniforme associado a área. / The aim of this work is analizing the process of integration of functions. There are many generalizations of the integration concept originally addressed by Riemann integral such as the Riemann-Stieltjes integral, Lebesgue integral, Henstock-Kurzweil integral, among others. We will be specially concerned with the integral of Riemann-Stieltjes and we will show the limitations of Riemann integral about convergence of functions, leading to the need to generalize the integration process. We will apply Riemann-Stieltjes integral for the study of random variables and present an approach to the classroom, on the displacement and distance traveled by an object in uniform rectilinear motion associated to concept of area.