Role of Simple Functions in the Lebesgue Integration I am re-reading the approximation by simple or step functions, and I don't quite understand what we are doing.
I get the idea that we are trying to approximate $f$, which is non-negative, measurable on $\mathbb{R}^d$. So we explore an increasing sequence of non-negative simple functions that converges pointwise to $f$.
But why is this important? Is there something analogous in the Riemann integration? Is there something similar to the simple function that we consider in the Lebesgue integration in the Riemann integral? If someone could explain the BIG picture, I would absolutely appreciate it.
Reference:
$\textit{Real Analysis: Measure Theory, Integration, and Hilbert Spaces}$. Elias M. Stein, Rami Shakarchi. Princeton University Press, 2009.
 A: The role of simple functions in Lebesgue integration is hard to over-emphasize. Essentially, we prove an enormous number of theorems by proving them first for simple functions, and then, because we have a number of theorems that let us generalize from simple functions to functions in general, we end up proving the theorem for functions in general. In fact, I remember my complex analysis professor talked about how he taught Real Analysis: students were generally allowed to prove something for simple functions, and then just write, "Building blocks." That meant they were invoking this idea that, because you can approximate any function "well enough" with simple functions, you just proved your result for all functions.
This, indeed, is the process of analysis in general: often a problem we want to solve is too difficult in its entirety. So what do we do? We break the problem down into smaller pieces, solve the smaller problem, and then figure out how to generalize the smaller solution to the whole problem. In analysis, as often as not, we just integrate to solve the whole problem.
