Comparison between Two Definitions of Lebesgue Integral

There are many methods to define Lebesgue integral. Here we compare the methods of Partition with ApproximationJust as Riemann integral, the former defines the Lebesgue integral by partitioning measurable sets and constructing Darboux's sum. It is very natural but unfavorable to bring out the key theories of Lebesgue integral. The latter defines the Lebesgue integral of a general function to be the limit if integrals of a sequence of simple functions that converges to it. The latter is less abstract, but it is advantageous to bring out three key theorems of Lebesgue integral.