After reading to this comments on this question which I asked about how to the actual caculation part of Lebesgue integrals differ from Riemann, I am quite confused on the usage of simple functions in itself. Why not just consider the region bounded by curves and two lines as a set to define the integral?
So, for eg, if I was thinking of the lebesgue integral of $\sin(x)$ on $\left[0,\frac{\pi}{2} \right]$, I consider the set:
$$ I = \{ (x,y) , 0<x<\frac{\pi}{2} , 0<y \leq sin(x)\}$$
Now just consider the measure of $I$ as a set, and define that as the lebesgue integral of sin on the interval as outer measure of it?
Why don't mathematicians just use this definition? Is this in some sense weaker than the way we do it through the simple functions?
Best Answer
Any integral you want to evaluate explicitly, has only 'one' way, namely the FTC. And the Lebesgue integral does that better than Riemann as well. So, $\int_{[0,\pi/2]}\sin x\,dm(x)=[-\cos(x)]_0^{\pi/2}=1$.
Anyway, one can define the integral (for non-negative functions) as the measure (not outer measure (but ok here you have a product-measurable set, so they're the same thing)) of the set. Charles Pugh's real analysis text does it this way (integral of non-negative functions defined as the measure of the undergraph). I personally find the simple functions approach quicker and also relatively intuitive, but if you find undergraphs nicer, then be my guest. But at the end of the day, after defining the integral, there's still the question of evaluating it explicitly. For that, you can't escape the FTC (which like I said above, Lebesgue does better).
Your comment reminded me of another fact. While we often teach (say Riemann) integrals as "area under the curve", and while this may (I don't know the precise history) have been the original motivation, the integral is so much more than that. The 'real' purpose of the integral is to generalize the idea of summation, and this is something the above approach fails to capture on a first pass. Sure, the above geometric approach may be nice, but can easily give a false impresssion of what integrals can/should do. For one, you can integrate vector-valued maps, as I mentioned in the link you commented. Next, see Visualization of double integrals.