Most books on Lebesgue integration define the concept first for step functions (and simple functions) and later on extend the definition to Lebesgue measurable functions. Why is this approach preferred over the original definition of Lebesgue which develops measure theory first and then defines the integral as a limit of sum (somewhat like the usual definition of Riemann integral)? Is there any pedagogic advantage? To me this approach looks quite artificial compared to the limit of a sum definition.
EDIT: In response to the comments from "Did" I am trying to elaborate further. I find two approaches available to study Lebesgue Integration:
1) Integral is defined for step functions using a simple sum. Then we consider increasing sequences of such step functions whose integrals converge to some value $I$. In this case the sequence of step functions converges a.e to some function $f$ and we say that $I$ is integral of $f$. Such functions as $f$ are called upper functions and difference of two such upper functions is called a Lebesgue Integrable function. This is the approach in Apostol's Mathematical Analysis. Concept of measure is defined later as integral of the indicator function. Royden's Real Analysis also follows the same approach.
2) Lebesgue's original definition where he defines the concept of measure first and then partitions the range of functions into multiple subintervals and forms a sum analogous to a Riemann sum and the limit of this sum as we make finer and finer partitions of the range is defined as the integral of the function.
I find almost all books to be using the first approach and not second. And Lebesgue's definition seems to be used only for historical context. To me the second approach looks intuitive. I wonder if there is any pedagogic advantage of the first approach. I am a beginner in these topics and still trying to come to terms with the step functions approach which is so desperately based on the convergence of sequence of functions.
Best Answer
The problem ist how to define the sum without step functions. The only way I could think of would use the density Function $$ d_f(t) := \mu(\{s\in D| f(s) > t\})$$ and the decreasing rearrangement. When you define the integral for step funcitons first however, you can prove lebesgue's convergence theorem and define the integral without such constructs. What limit would you use to define the integral of an arbitrary function without using an approximation with step functions?