# Henstock-Kurzweil Integration Theory

General information

The Henstock-Kurzweil integral was introduced in 1957-62 by Ralph Henstock and independently by Jaroslav Kurzweil. The H-K integral gives a Riemannian definition for the Perron and restricted Denjoy integrals. Thus the H-K integral encompasses the integrals of Newton, Riemann e Lebesgue as well as its improper integrals.

Also known as the Generalized Riemann Integral or the Complete Riemann Integral, the H-K integral has good convergence and functional analytic properties. Moreover, its definition can be easily extended to functions defined on unbounded intervals and Banach space-valued.

The main feature of the non-absolute H-K integral is the fact that functions with not anly many discontinuities, but also highly oscillating can be integrable in Henstiock-Kurzweil sense. For instance, the function f defined on the compact interval [0,1] of the real line and given by f(t)=F'(t), where F(t)=t^2sen(1/t^2) on ]0,1] and F(0)=0 is H-K integrable (but not Lebesgue nor Riemann integrable).

With a slight modification in its definition, the H-K integral gives way to a more restricted integral known as the McShane integral which coincides with the Lebesgue integral. Thus, we have a constructive definition for the Lebesgue integral by means of Riemann sums. This fact enables one to apply the techniques and tools of the Henstock-Kurzweil Integration Theory to treat problems which may involve Lebesgue integrable functions (with many discontinuities, for instance). In this manner, this theory has been shown to be useful in the more modern treatment of integral and differential equations centered in weak solutions (almost everywhere differentiable, for instante).

Some textbooks in this area

• Basic textbooks:
• R. G. BARTLE e D. R. SHERBERT, Introduction to Real Analysis, John Wiley and Sons, Inc., 2000, 3a Edição.
• J. DEPREE e C. SWARTZ, Introduction to Real Analysis, John Wiley and Sons Inc., 1988.
• R. A. GORDON, The Integrals of Lebesgue, Denjoy, Perron, and Henstock, Graduate Studies in Math. v. 4, American Mathematical Society, 1994.
• C. S. HÖNIG, As Integrais de Gauge, Minicurso, Seminario Brasileiro de Análise, v. 37, 1993.
• R. M. MC LEOD, The Generalized Riemann Integral, Carus Math. Monog., 20, The Math. Ass. of America, 1980.
• E. SCHECHTER, Handbook of Analysis and its Foundations.