A cauchy criterion and a convergence theorem for Riemann-complete integral

In 1957 Kurzweil [1] proved some theorems concerning a generalized type of differential equations by defining a Riemann-type integral, but he did not study its properties beyond the needs of that research. This was done by R. Henstock [2, 3], who named it a Riemann-complete integral. He showed that the Riemann-complete integral includes the Lebesgue integral and that the Levi monotone convergence theorem holds. The purpose of the present paper is to give a necessary and sufficient condition for a function to be Riemann-complete integrable and to establish a termwise integration theorem for a uniformly convergent sequence of Riemann-complete integrable functions.