The Ross-Darboux-Stieltjes integral
DOI:
https://doi.org/10.12775/TMNA.2025.058Słowa kluczowe
Stieltjes integral, Lebesgue-Stieltjes integral, functions of bounded variationAbstrakt
Motivated by the limitations of the traditional definitions of the Riemann-Stieltjes and Darboux-Stieltjes integrals, we introduce a generalized Darboux-Stieltjes integral that is equivalent to an earlier generalization by Ross. Our definition builds upon an approach to the Darboux-Stieltjes integral recently introduced by the first author and Convertito in their recent book. We show that our definition agrees with all previous definitions, but that the class of integrable functions is much larger. We develop all the analogs of the classic results for the Riemann integral, and rectify the problems inherent in the definition of the Darboux-Stieltjes integral in this book. In particular, we show the bounded convergence theorem holds for our definition and that it agrees with the Lebesgue-Stieltjes integral.Bibliografia
T.M. Apostol, Mathematical Analysis: a Modern Approach to Advanced Calculus, Addison-Wesley Publishing Company, Inc., Reading, Mass., 1957.
J. Appell, J. Banaś and N. Merentes, Bounded Variation and Around, De Gruyter Series in Nonlinear Analysis and Applications, vol. 17, De Gruyter, Berlin, 2014.
R.G. Bartle, The elements of real analysis, John Wiley & Sons, New York, London, Sydney, second edition, 1976.
J.C. Burkill and H. Burkill, A second Course in Mathematical Analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2002, reprint of the 1970 original.
M. Carter and B. van Brunt, The Lebesgue–Stieltjes Integral: a Practical Introduction, Undergraduate Texts in Mathematics, Springer–Verlag, New York, 2000.
G. Convertito and D. Cruz-Uribe, The Stieltjes Integral, CRC Press, Boca Raton, FL, 2023.
R.M. Dudley and R. Norvaiša, Concrete Functional Calculus, Springer Monographs in Mathematics, Springer, New York, 2011.
B. Dushnik, On the Stieltjes Integral, ProQuest LLC, Ann Arbor, MI, 1931, Ph.D. Thesis, University of Michigan.
E. Hewitt and K. Stromberg, Real and Abstract Analysis. A Modern Treatment of the Theory of Functions of a Real Variable, Springer–Verlag, New York, 1965.
T.H. Hildebrandt, Definitions of Stieltjes integrals of the Riemann type, Amer. Math.
H.J.T. Horst, Riemann–Stieltjes and Lebesgue–Stieltjes integrability Amer. Math. Monthly 91 (1984), no. 9, 551–559.
E. Kamke, Das Lebesgue–Stieltjes-Integral, B.G. Teubner Verlagsgesellschaft, Leipzig, 1956.
S.G. Krantz, Real Analysis and Foundations, Textbooks in Mathematics, CRC Press, Boca Raton, FL, fourth edition, 2017.
W.A.J. Luxemburg, Arzelà’s dominated convergence theorem for the Riemann integral, Amer. Math. Monthly 78 (1971), 970–979.
G.A. Monteiro, A. Slavı́k and M. Tvrdý, Kurzweil–Stieltjes Integral, Series in Real Analysis, vol. 15, Theory and Applications, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2019.
I.P. Natanson, Theory of functions of a real variable, Frederick Ungar Publishing Co., New York, 1955, translated by Leo F. Boron with the collaboration of Edwin Hewitt.
O.A. Nielsen, An Introduction to Integration and Measure Theory, Canadian Mathematical Society Series of Monographs and Advanced Texts, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1997.
S. Pollard, The Stieltjes’ integral and its generalisations, Quart. J. 49 (2021), 73–138.
M.H. Protter and C.B. Morrey A First Course in Real Analysis, Undergraduate Texts in Mathematics, Springer–Verlag, New York, Berlin, 1977.
F. Riesz and B. Sz.-Nagy, Functional Analysis, Frederick Ungar Publishing Co., New York, 1955. Translated by Leo F. Boron.
M. Riesz, Sur les opérations fonctionelles linéaires, Comptes Rendus Mathématique. Académie des Sciences, Paris, 149 (1909), 974–977.
K.A. Ross, Another approach to Riemann–Stieltjes integrals, Amer. Math. Monthly 87 (1980), no. 8, 660–662.
K.A. Ross, Elementary Analysis: the Theory of Calculus, Undergraduate Texts in Mathematics, Springer–Verlag, New York, Heidelberg, 1980.
H.L. Royden, Real Analysis, Macmillan Publishing Company, New York, third edition, 1988.
W. Rudin, Principles of Mathematical Analysis, International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York, Auckland, Düsseldorf, third edition, 1976.
T.-J. Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse Math. 8 (1894), no. 4, 1–122.
A. Taylor and W. Mann, Advanced Calculus, k John Wiley & Sons, Inc., New York, third edition, 1983.
R.L. Wheeden and A. Zygmund, Measure and Integral, Pure and Applied Mathematics (Boca Raton), second edition, CRC Press, Boca Raton, FL, 2015.
Pobrania
Opublikowane
Jak cytować
Numer
Dział
Statystyki
Liczba wyświetleń i pobrań: 0
Liczba cytowań: 0
