The text is a self-contained account of integration theory on the real line. The usual curricula in real analysis courses do not allow for much time to be spent on the Henstock-Kurzweil integral. Instead extensive accounts of Riemann’s integral and the Lebesgue integral are presented. Accordingly the version here would be mostly recommended for supplementary reading.

Even so it would be a reasonable course design to teach this material prior to a course in abstract measure and integration. The student should end up as well-prepared as in more traditional courses. Certainly every professional mathematician should be aware of more than Lebesgue’s theory; while nonabsolutely convergent integrals do not play an extensive role in applications, they are part of our history and of our culture.

There are innumerable books published on the Lebesgue integral. The reader needing instruction in that theory is faced with too many choices, although many of them are truly excellent. For the more general integral (called here the general Newton integral or simply “the integral”) that is best known classically as the Denjoy-Perron integral and, more recently, as the HenstockKurzweil integral, there are far fewer choices and not all of them are excellent. I have resisted for many years writing a lengthy account of this integral, partly because the topic is not widely thought of as being of much significance.