A Deep Dive Into Integrals: From Riemann, Stieltjes, Lebesgue to Henstock–Kurzweil

Integration theory is broadly concerned with the existence of solutions to definite integrals over closed subintervals of the real line, and its power lies in the various techniques of integration, which enable integration of successively broader classes of functions. In this work, we establish a self–contained theory of integration in the sense of Riemann, Riemann–Stieltjes, Lebesgue, and Henstock–Kurzweil. Notably, our work utilizes the theory of Lebesgue integration developed by P. J. Daniell, which does not rely on measure theory and thus remains accessible at the undergraduate level. Our incentive for such an approach is to improve equitable access to advanced integration theory in undergraduate learning environments. This work remains grounded in the applications of integration theory, which span the disciplines of probability theory, complex analysis, quaternionic theory, and even quantum mechanics. Finally, we present the Vitali set indicator function and a novel proof that this function is not integrable in the sense of any of the integration techniques surveyed, thus crystallizing the limits of integration theory as it exists today. i