The definitions of Lebesgue integral of non-negative measurable functions and their equivalence

The theorem of Lebesgue integral is one of the most important parts in real variable function. About Lebesgue integral,there are different definition modes.For simplifying and unifying the defini- tions,and for understanding and mastering the concepts of Lebesgue integral from different angle,it is very significant to study how to define Lebesgue integral,and to prove the equivalence of different defin- tions.In this paper,the definition modes of Lebesgue integral of non-negative measurable functions are studied by the way of cutting defining and valued rids,approaching with simple function series.The four definitions of Lebesgue integral of non-negative measurable functions are given.Furthermore,their e- quivalence properties are proved by elementary knowledge.