Note that it is recommended that the reader have some knowledge of the Riemann and Lebesgue integrals.
In the same way that the Lebesgue integral generalizes the Riemann integral, the Henstock-Kurzweil integral (HK integral) generalizes the Lebesgue integral. Yet it does so without requiring the deep theorems of measure theory. Instead, the definition of the HK integral is remarkably similar to the Riemann integral.
This integral first appeared in the forms described by Denjoy and by Perron. These turned out to be equivalent, but it was hard to follow their formulations. Later Henstock and Kurzweil simplified the description of this integral and invented the Gauge integral. This new formulation was so simple that some educators advocate that this integral should replace the Riemann integral in introductory calculus courses.