The Relationship between Complete Convergence and Some Convergence of Measurable Function

The paper discusses the equivalent relationship between complete convergence and convergence in measure,almost everywhere convergence,strong convergence of fathomable functional sequence,and presentes a necessary and sufficient condition when convergence in measure and almost everywhere convergence are equivalent to complete convergence,namely,fn(s) monotone increasing,and(An) pairwise disjoint,where An=[|fn-f|≥e],e0.