A.7. SOME OBSERVATIONS 869

A.7 Some ObservationsSome of the above material is very technical. This is because it gives complete answersto the fundamental questions on existence of the integral and related theoretical considera-tions. However, most of the difficulties are artifacts. They should not even be considered!It was realized early in the twentieth century that these difficulties occur because, fromthe point of view of mathematics, this is not the right way to define an integral! Betterresults are obtained much more easily using the Lebesgue integral. Many of the tech-nicalities related to Jordan content disappear almost magically when the right integral isused. However, the Lebesgue integral is more abstract than the Riemannn integral and itis not traditional to consider it in a beginning calculus course. If you are interested in thefundamental properties of the integral and the theory behind it, you should abandon theRiemannn integral which is an antiquated relic and begin to study the integral of the lastcentury. An introduction to it is in [31]. Another very good source is [16]. This advancedcalculus text does everything in terms of the Lebesgue integral and never bothers to strug-gle with the inferior Riemannn integral. A more general treatment is found in [26], [27],[32], and [28]. There is also a still more general integral called the generalized Riemannnintegral. A recent book on this subject is [5]. It is far easier to define than the Lebesgueintegral but the convergence theorems are much harder to prove. An introduction is also in[27].