The widest continuous integral

An approach to a definition of an integral, which differs from definitions of Lebesgue and Henstock– Kurzweil integrals, is considered.We use trigonometrical polynomials instead of simple functions. Let V be the space of all complex trigonometrical polynomials without the free term. The definition of a continuous integral on the space V is introduced. All continuous integrals are described in terms of norms on V . The existence of the widest continuous integral is proved, the explicit form of its norm is obtained and it is proved that this norm is equivalent to the Alexiewicz norm. It is shown that the widest continuous integral is wider than the Lebesgue integral. An analog of the fundamental theorem of calculus for the widest continuous integral is given. © 2006 Elsevier Inc. All rights reserved.