A Crash Course on the Lebesgue Integral and Measure Theory

This booklet is an exposition on the Lebesgue integral. I originally started it as a set of notes consolidating what I had learned on on Lebesgue integration theory, and published them in case somebody else may find them useful. Since there are already countless books on measure theory and integration written by professional mathematicians, that teach the same things on the basic level, you may be wondering why you should be reading this particular one, considering that it is so blatantly informally written. Ease of reading. Actually, I believe the informality to be quite appropriate, and integral — pun intended — to this work. For me, this booklet is also an experiment to write an engaging, easy-to-digest mathematical work that people would want to read in my spare time. I remember, once in my second year of university, after my professor off-handed mentioned Lebesgue integration as a " nicer theory " than the Riemann integration we had been learning, I dashed off to the library eager to learn more. The books I had found there, however, were all fixated on the stiff, abstract theory — which, unsurprisingly, was impenetrable for a wide-eyed second-year student flipping through books in his spare time. I still wonder if other budding mathematics students experience the same disappointment that I did. If so, I would like this book to be a partial remedy. Motivational. I also find that the presentation in many of the mathematics books I encounter could be better, or at least, they are not to my taste. Many are written with hardly any motivating examples or applications. For example, it is evident that the concepts introduced in linear functional analysis have something to do with problems arising in mathematical physics, but " pure " mathematical works on the subject too often tend to hide these origins and applications. Perhaps, they may be obvious to the learned reader, but not always for the student who is only starting his exploration of the diverse areas of mathematics. iv 0. Preface: Special thanks v Rigor. On the other hand, in this work I do not want to go to the other extreme, which is the tendency for some applied mathematics books to be unabashedly un-rigorous. Or worse, pure deception: they present arguments with unstated assumptions , and impress on students, by naked authority, that everything they present is perfectly correct. Needless to say, I do …