The Riemann integral was introduced in Bernhard Riemann's paper "Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe" (On the representability of a function by a trigonometric series; i.e., when can a function be represented by a trigonometric series). This paper was submitted to the University of Göttingen in 1854 as Riemann's Habilitationsschrift (qualification to become an instructor). It was published in 1868 in Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen (Proceedings of the Royal Philosophical Society at Göttingen), vol. 13, pages 87-132. (Available online here.) For Riemann's definition of his integral, see section 4, "Über den Begriff eines bestimmten Integrals und den Umfang seiner Gültigkeit" (On the concept of a definite integral and the extent of its validity), pages 101–103.
Basic real analysis, by Houshang H. Sohrab, section 7.3, Sets of Measure Zero and Lebesgue’s Integrability Condition, pp. 264–271
doi.org
Brown, A. B. (September 1936). "A Proof of the Lebesgue Condition for Riemann Integrability". The American Mathematical Monthly. 43 (7): 396–398. doi:10.2307/2301737. ISSN0002-9890. JSTOR2301737.
Brown, A. B. (September 1936). "A Proof of the Lebesgue Condition for Riemann Integrability". The American Mathematical Monthly. 43 (7): 396–398. doi:10.2307/2301737. ISSN0002-9890. JSTOR2301737.
Introduction to Real Analysis, updated April 2010, William F. Trench, 3.5 "A More Advanced Look at the Existence of the Proper Riemann Integral", pp. 171–177
unapologetic.wordpress.com
Lebesgue’s Condition, John Armstrong, December 15, 2009, The Unapologetic Mathematician
Brown, A. B. (September 1936). "A Proof of the Lebesgue Condition for Riemann Integrability". The American Mathematical Monthly. 43 (7): 396–398. doi:10.2307/2301737. ISSN0002-9890. JSTOR2301737.
