On Generalized Non-ordinary Riemann Integrals

A new generalized non-ordinary Riemann integral is introduced and some properties of the convergence and divergence for the integral are discussed. It was proved that the convergence and the absolute convergence of the integral for a function on (-∞,+∞) are equivalent, and that if the integral of a function f(x) converges to I, then f(x) is integrable in the sense of Lebesgue and its Lebesgue integral has the same value I.