| Abstract: | The theory of finding, he limit under Riemann integral sign is to search the conditions such that the equality
■integral from n=a to b f_n(x)dx=integral from n=a to b■f_n(x)]dx
holds. The general conditions in mathematical analysis are as follows:
1)f_n(x)∈Ca, b], (n =1, 2, ");
2)f_n(x)(■f(x).
These conditions are so strong that they are not satisfied for many ordinary examples, but the limits can be found under Riemann integral sign.
In this paper, two fundamental concepts and a lemma are introduced. By means of these and the theory of measure and Lebesgue integral, We first establish a more general proponsition(Th. 1)which weakens the above conditions. At last we give the analogues(Th. 2, Th. 3 and Corollaries) for improper integals. |
