Abstract: The application of Lebesgue dominated convergence theorem and the necessary and sufficient condition of strong convergence are discussed in this paper.New simplified proof methods of some classic results are obtained by means of recent results derived by Lebesgue dominated convergence theorem and the discriminant theorem of the necessary and sufficient condition of strong convergence is demonstrated.

Key words: Lebesgue dominated convergence theorem, strong convergence, almost sure convergence, convergence in measure, weak convergence