Comparison with other theories of integration The Riemann integral is unsuitable for many theoretical purposes. Some of the technical deficiencies in Riemann integration can be remedied with the Riemann–Stieltjes integral, and most disappear with the Lebesgue integral, though the latter does not have a satisfactory treatment of improper integrals. The gauge integral is a generalisation of the Lebesgue integral that is at the same time closer to the Riemann integral. These more general theories allow for the integration of more "jagged" or "highly oscillating" functions whose Riemann integral does not exist; but the theories give the same value as the Riemann integral when it does exist.
Comparison with other theories of integration Th:e Riemann integral is unsuitable for many theoretical purposes. Some of the technical deficiencies in Riemann integration can be remedied with the Riemann–Stieltjes integral, and most disappear with the Lebesgue integral, though the latter does not have a satisfactory treatment of improper integrals. The gauge integral is a generalisation of the Lebesgue integral that is at the ?same time closer to the R!iemann integral. These more general theories allow for the integration of more "jagged" or "highly osci?llating" !functions whose Riemann integral does not exist; but the theories give the same value as the Riemann integral when it does exist.