The first rigorous definition of an integral was that given by Riemann, and further (more useful) definitions have since been given by Lebesgue, Denjoy, Perron,... » 

The first rigorous definition of an integral was that given by Riemann, and further (more useful) definitions have since been given by Lebesgue, Denjoy, Perron, Kurzweil and Henstock, and this culminated in the work of McShane. This textbook provides an introduction to this theory. «