An Alternate Approach to Integration
We can wrap up this introduction to the Lebesgue integral by outlining the alternate approach that commenter Cristi was referring to. We’ll do this from the perspective of our current track, and it should be clear how the alternative definitions would lead us to the same place.
If is a non-negative integrable function on a measure space . For every measurable set , we define
Also, for every finite, pairwise disjoint collection of measurable sets we define
We then assert that the supremum of all numbers for all finite, pairwise disjoint collections is equal to the integral of :
If is simple, this is obvious. Indeed, if is the collection of sets used to write as a finite linear combination of characteristic functions, then is exactly the integral of by definition. Any set that extends outside one of these sets will have , and so we can’t get any larger than the integral of .
On the other hand, for a general integrable function we consider a non-negative simple with , and we let be the sets used to express as a finite linear combination of characteristic functions:
We see that
If is an increasing sequence of non-negative simple functions converging pointwise a.e. to , then
where we use the definition of integrability, and we take the supremum over finite, pairwise disjoint collections . But it’s also clear that for every we have
for some non-negative simple .
So the alternate approach proceeds by defining the integral of a simple function as before, and defining general integrals of non-negative functions by the supremum above. General integrable functions overall are handled by using their positive and negative parts. Then you can prove the monotone convergence theorem, followed by Fatou’s lemma, and then the Fatou-Lebesgue theorem, which leads to dominated convergence theorem, and we’re pretty much back where we started.
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