On Extension Methods of Uniform Convergence and Their Interconnections

This paper gives a description of the set sequences,which satisfies nearly uniform convergence,almost everywhere convergence,or convergence in measure.The application of a collection of sequence greatly simplifies the proof of series theorems such as known EropoB and Lebesgue theorem.It expands the Riesz Theorem fromstudying the relationship of convergence in measure and almost everywhere convergencetostudying the relationship between convergence in measure and near uniform convergence,and breaks through the condition of mE+∞, proves inverse theorem of Riesz theorem after the Extension.