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A Generalization of the Riemann Integral [PDF]
Proceedings of the American Mathematical Society, 1952 By considering certain kinds of sets as exceptional in the definition of the Riemann integral, a variety of possible generalizations of the integral concept for bounded functions is obtained, one of which is the Lebesgue integral. Such an extension has been discussed by E. H.
Casper Goffman
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Pattern Integration with Improper Riemann Integrals [PDF]
Proceedings of the American Mathematical Society, 1951 2. The improper integral defined as the limit of a sum. We first consider the case where ak = 1 (k =1, 2, 3, * * * ). A fundamental difficulty not encountered in the previous paper is apparent from the definition of .fof(x)dx. Bromwich and Hardy have considered this problem [2]. As they point out, the ordinary definition of this improper integral is by
R. E. Carr
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The Lebesgue integral as a Riemann integral [PDF]
International Journal of Mathematics and Mathematical Sciences, 1987 The object of this paper is to develop a very direct theory of the Lebesgue integral that is easily accessible to any audience familiar with the Riemann integral.
Enrique A. Gonzalez-Velasco
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ON THE POSITIVITY OF RIEMANN–STIELTJES INTEGRALS [PDF]
Bulletin of the Australian Mathematical Society, 2012 AbstractWe study the question whether a Riemann–Stieltjes integral of a positive continuous function with respect to a nonnegative function of bounded variation is positive.
Lukkarinen, Jani, Pakkanen, Mikko S.
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Existence and computation of Riemann-Stieltjes integrals through Riemann integrals
, 2011 We study the existence of Riemann-Stieltjes integrals of bounded functions against a given integrator. We are also concerned with the possibility of computing the resulting integrals by means of related Riemann integrals. In particular, we present a new generalization to the well-known formula for continuous integrands and continuously differentiable ...
Rodrigo López Pouso
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On the Riemann-Stieltjes Integral [PDF]
Mathematics Interdisciplinary Research, 2022 This study contributes to the theory of Riemann-Stieltjes integral. We prove that if all continuous piecewise linear functions are Riemann-Stieltjes integrable with respect to a bounded integrator α : [a,b] → R, then α must be of bounded variation on [a ...
Ali Parsian
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On interval-valued $\mathbb{K}$-Riemann integral and Hermite-Hadamard type inequalities
AIMS Mathematics, 2021 We introduce the concepts of $\mathbb{K}$-Riemann integral and radial $\mathbb{K}$-g$H$-derivative for interval-valued functions. We also give some important properties of interval-valued $\mathbb{K}$-Riemann integral, and extend interval-valued Hermite ...
Zehao Sha +3 more
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AIMS Mathematics, 2021 In the present note, we develop Hermite-Hadamard type inequality and He's inequality for exponential type convex fuzzy interval-valued functions via fuzzy Riemann-Liouville fractional integral and fuzzy He's fractional integral.
Yanping Yang +3 more
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Riemann integral and its relation with Lebesgue integral
Bibechana, 2010 This review explains the limitations of the Riemann integral and focuses on the more general concept. It has been suggested that a better route is to abandon the Riemann integral for Lebesgue integral and explained the importance of Lebesgue integral in ...
Mani Raj Bhattrai
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A Comprehensive Review on the Fejér-Type Inequality Pertaining to Fractional Integral Operators
Axioms, 2023 A review of the results on the fractional Fejér-type inequalities, associated with different families of convexities and different kinds of fractional integrals, is presented.
Muhammad Tariq +2 more
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