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.

La méthode des régions-solutions a été développée par Frigon [7] en 2018 pour montrer l'existence de solutions à des équations différentielles ordinaires de premier ordre, dont le graphe d'une solution se trouve à l'intérieur d'une région-solution \(R \subset [0, T] \times \mathbb{R}^{N}\). Cette méthode est en particulier une généralisation des sous et sur-solutions et des tubes-solutions. On présente cette méthode et certains résultats d'existence qui en découlent. D'autre part, la dérivée de Stieltjes, communément appelée \(g\)-dérivée, est le fruit du travail de Pouso et Rodríguez [20] en 2014, permettant l'unification des équations différentielles classiques, des équations aux échelles de temps et des équations différentielles avec impulsions. Elle est en particulier liée au théorème fondamental du calcul pour l'intégrale de Lebesgue-Stieltjes. On présente la base de cette théorie dans un premier temps, puis la façon dont cette \(g\)-dérivée généralise d'autres types d'équations différentielles ou aux échelles de temps. On introduit en particulier la notion de \((g \times I_{\mathbb{R}^{N}})\)-différentiabilité et des résultats qui découlent de cette définition. On présente de plus une fonction exponentielle qui permet de résoudre les équations différentielles de Stieltjes linéaires, introduite par Frigon et Pouso [8]. Le but de ce mémoire est de généraliser la méthode des régions-solutions, dont la généralisation s'appellera \(g\)-région-solution, afin de montrer l'existence de solutions aux équations différentielles de Stieltjes. On présente plusieurs exemples de \(g\)-régions admissibles et de \(g\)-régions-solutions, puis des théorèmes d'existence se basant sur cette méthode. On donne de plus des exemples où on applique ces théorèmes. On termine ce mémoire en présentant deux applications des théorèmes d'existence à l'évolution d'une population de cerfs de Virginie ainsi qu'à l'évolution de la tension générée par une diode à effet tunnel résonnant (DTR) dirigée vers une diode laser (DL). / The method of solution-regions has been developed by Frigon [7] in 2018 to show the existence of solutions for first-order ordinary differential equations, where the graph of a solution is inside a solution-region \(R \subset [0, T] \times \mathbb{R}^{N}\). This method is in particular a generalization of the lower and upper solutions and of the solution-tubes. We show this method and some existence results which follow. On the other hand, the Stieltjes derivative, more commonly called \(g\)-derivative, is the fruit of the work of Pouso and Rodríguez [20] in 2014, which unifies classic differential equations, equations on time scales and differential equations with impulses. In particular, it leads to the fundamental theorem of calculus for the Lebesgue-Stieltjes integral. We start by showing the basis of this theory, and then the way this \(g\)-derivative generalizes other types of differential or time scale equations. We introduce in particular the \((g \times I_{\mathbb{R}^{N}})\)-differentiability and results that follow from this definition. Furthermore, we present an exponential function which solves linear Stieltjes differential equations. The goal of this thesis is to generalize the method of solution-regions, where the generalization will be called \(g\)-solution-region, to show the existence of solutions for Stieltjes differential equations. We present multiple examples of \(g\)-admissible regions and \(g\)-solution-regions, then we establish existence theorems based on this method. We also show examples where we apply these theorems. Finally, we end this thesis by showing two applications of the existence theorems to the evolution of a population of white-tailed deer and to the evolution of the voltage generated by a resonant tunneling diode (RTD) to a laser diode (LD).

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)