Jal Choksi - A history of the convergence theorems of (Lebesgue) integration

In most courses on Lebesgue integration there are three main convergence results: (i)  the monotone convergence theorem (MCT), (ii)  Fatou's lemma, both for non-negative functions, (iii)  the dominated convergence theorem (DCT) and its corollary (for finite total measure) the bounded convergence theorem (BCT). They are most often proved in that order. DCT is the most often used in practice. Historically, things were very different! Lebesgue's thesis (1902) contains only BCT. It was only 4 years later, in 1906, that Beppo Levi proved MCT and independently Fatou proved his lemma. The proofs, each starting with BCT, are very similar. DCT first appears in a paper of Lebesgue in 1908, with a more detailed account in a paper in 1910. The proof is similar to his original proof of BCT. Earlier, in 1907, Vitali had proved a convergence theorem using the concept of uniform absolute continuity of the integrals, we shall discuss this and it's subsequent use. We shall also talk about the work of F. Riesz and Fischer on L² convergence (starting around 1905), and Riesz' subsequent generalization to L^p, but this history is better known. The best reference book is Hawkins, Lebesgue's theory of integration, but our lecture may contain a few surprises, even to those who have read this book!