Necessary and Sufficient Conditions for Riemann and Riemann-stieltjes Integrability
ثبت نشده
چکیده
RIEMANN AND RIEMANN-STIELTJES INTEGRABILITY Looking back over the notes already posted on this and related subjects, I felt that their organization did not emphasize the things that are important on the subject of the caption. Hence these notes, which will initially reprise things that are already treated in the others, but with different emphasis. The material on almost-everywhere continuity as a necessary and sufficient condition for integrability will be new, in the sense that both my notes and the textbook rely on Lebesgue(-Stieltjes) integration to establish this condition, while these notes will give a completely elementary treatment. 1. Integrability Characterized by a Limit Condition. Theorem: In order that ∫ b
منابع مشابه
Report for MA502 Presentation: Lebesgue’s Criterion for Riemann Integrability
We covered Riemann integrals in the first three weeks in MA502 this semester (Chapter 11 in [1]). This report explores a necessary and sufficient condition for determining Riemann integrability of f(x) solely from its properties. This condition is known as Lebesgue’s criterion and elucidating the proof of this condition is the aim of this report. This report revisits the concepts learnt in MA50...
متن کاملNvestigation of a Boundary Layer Problem for Perturbed Cauchy-Riemann Equation with Non-local Boundary Condition
Boundary layer problems (Singular perturbation problems) more have been applied for ordinary differential equations. While this theory for partial differential equations have many applications in several fields of physics and engineering. Because of complexity of limit and boundary behavior of the solutions of partial differential equations these problems considered less than ordinary case. In ...
متن کاملRiemann-Stieltjes integrals
This short note gives an introduction to the Riemann-Stieltjes integral on R and Rn. Some natural and important applications in probability theory are discussed. The reason for discussing the Riemann-Stieltjes integral instead of the more general Lebesgue and LebesgueStieltjes integrals are that most applications in elementary probability theory are satisfactorily covered by the Riemann-Stieltj...
متن کاملan 2 00 1 Necessary and sufficient conditions for differentiating under the integral sign
a f(x, y) dy, it is often important to know when F is differentiable and when F (x) = ∫ b a f1(x, y) dy. A sufficient condition for differentiating under the integral sign is that ∫ b a f1(x, y) dy converges uniformly; see [6, p. 260]. When we have absolute convergence, the condition |f1(x, y)| ≤ g(y) with ∫ b a g(y) dy < ∞ suffices (Weierstrass M-test and Lebesgue Dominated Convergence). If we...
متن کاملA Note on the Convergence of Random Riemann and Riemann-Stieltjes Sums to the Integral
Convergence in probability of random Riemann sums of a Lebesgue integrable function on [0, 1) to the integral has been proved. In this article we generalize the result to an abstract probability space under some natural conditions and we show L1convergence rather than convergence in probability..
متن کامل