The dominated convergence theorem for the Riemann and the improper Riemann integral (Measure theory is a must – part II)

(Hello students of Infi 2 – this post is for you).

In this post I will describe the dominated convergence theorem (DCT) for the Riemann and improper Riemann integrals. The previous post can serve as an introduction (a slanted one, beware) to this one. My goal is to convince that the important and useful convergence theorems in integration theory can (and therefore, needless to say, should) be taught in a first course on Riemannian integration.

The bounded convergence theorem for the Riemann integral is also known as Arzela’s Theorem, and this post does not contain anything new. In preparing this post I used as reference the short note “A truly elementary approach to the bounded convergence theorem”, J. W. Lewin, The American Mathematical Monthly. This post can be considered as a destreamlinization of that note. I think my presentation is even more “truly elementary”, since I avoid introducing inner measure. Warning: this post will really truly be at a very elementary level.

Measure theory is a must

[This post started out as an introduction to a post I was planning to write on convergence theorems for the Riemann integral. The introduction kind of got out hand, so I decided to post it separately. Since I have to get back to my real work, I will postpone writing that post on convergence theorems for the Riemann integral for another time, probably during the Passover break (but in any case before we need them for the course I am teaching this term, Calculus 2)].

 

Mathematicians love to argue about subjective opinions. One of the most tiresome and depressing subjects of debate is “What should an undergraduate math major curriculum contain?”