Historical Developments

Lebesgue begins his exposition by briefly mentioning the classical theorems that underpin the analysis of singular integrals, assuming that his readers are already familiar with these foundational concepts. He then builds upon this knowledge, developing his own analysis of singular integrals. In many ways, this work serves as a continuation of, and a response to, the work of mathematicians such as Dirichlet, Du Bois-Reymond, Dini, Jordan, and E.W. Hobson. While Lebesgue’s work is self-contained, we believe it is valuable to first consolidate the relevant theorems that provide the basis for his contributions, which we do in this chapter. These foundational theorems and background material can be found in various resources, referenced where appropriate. Before examining the convergence of certain types of integrals, it is both interesting and essential to consider the underlying motivations that led to their study.