A question on the integral of nonnegative functions using the integral of simple functions

Question

Question: In Folland's Real Analysis, on page $50$, he says, when discussing the integral of simple functions, that we can extend the integral of a simple function to all nonnegative functions $f$ by defining $$\int fd\mu=\sup\{\int \phi d\mu :0\leq \phi \leq f, \phi\text{ simple}\}$$

I have two questions: First, why can we do this? Is it because for all nonnegative measurable functions $f$, there is a sequence of nondecreasing simple functions which are all $\leq f$ such that they converge pointwise to $f$ (and converge uniformly to $f$ is $f$ is on a bounded set)? Second, could someone provide an example with an explicit simple function and function $f$? I suppose I "get" the definition, but I am just wondering what this actually looks like for a "practical" example.

Thank you!

Answer 1

The reason it is a sensible definition is because we can approximate nonnegative measurable functions from below by simple functions. Practical examples rarely appeal explicitly to the supremum, and more commonly use convergence theorems to compute integrals. Some practical examples are:

1. $\int 1_A \,d\mu = \mu(A)$
2. Lebesgue integral on $\mathbb{R}^n$ coincides with Riemann's integral (hence FTC is applicable).
3. Integration with counting measure is summation.
4. Expectation of a random variable is integration. This is nothing more than a restatement of the definition of expectation, but expectation somehow has a different flavor than integration.