Equiintegrability and controlled convergence for the Henstock-Kurzweil integral

Why we need Non absolute integral in place of Lebesgue integral?

“…While a substantial amount of the theory of the Lebesgue integral can be developed from the Henstock-Kurzweil integral, it is likely that the Henstock-Kurzweil integral has to contribute something highly practical which the Lebesgue integral does not, in order for it to have a chance of at least partially replacing the Lebesgue integral in advanced analysis courses. Some research involving the Henstock-Kurzweil integral, partial differential equations and integral transformations has been done, see for example (Mema, 2013;Mohanty and Talvila, 2003). Sometimes we are interested in solving problems with minimal smoothness assumptions and the solutions to such problems might involve bad functions that the theory of Lebesgue cannot deal with.…”