Cfrgtkky

In order to define Lebesgue integral, we have to develop some measure theory. This takes some effort in the classroom, after which we need additional effort of defining Lebesgue integral (which also adds a layer of complexity). Why do we do it this way?

The first question is to what extent are the notions different. I believe that a bounded measurable function can have a non-measurable "area under graph" (it should be doable by transfinite induction), but I am not completely sure, so treat it as a part of my question. (EDIT: I was very wrong. The two notions coincide and the argument is very straightforward, see Nik Weaver's answer and one of the comments).

What are the advantages of the Lebesgue integration over area-under-graph integration? I believe that behaviour under limits may be indeed worse. Is it indeed the main reason? Or maybe we could develop integration with this alternative approach?

Note that if a non-negative function has a measurable area under graph, then the area under the graph is the same as the Lebesgue integral by Fubini's theorem, so the two integrals shouldn't behave very differently.

EDIT: I see that my question might be poorly worded. By "area under the graph", I mean the measure of the set of points $(x,y) in E times mathbb{R}$ where $E$ is a space with measure and $y leq f(x)$. I assume that $f$ is non-negative, but this is also assumed in the standard definition of the Lebesuge integral. We extend this to arbitrary function by looking at the positive and the negative part separately.

The motivation for my question concerns mostly teaching. It seems that the struggle to define measurable functions, understand their behaviour, etc. might be really alleviated if directly after defining measure, we define integral without introducing any additional notions.

Actually, in the following book the Lebesgue integral is defined the way you suggested:

Pugh, C. C. Real mathematical analysis. Second edition. Undergraduate Texts in Mathematics. Springer, Cham, 2015.

First we define the planar Lebesgue measure $m_2$. Then we define the Lebesgue integral as follows:

Definition. The undergraph of $f:mathbb{R}to[0,infty)$ is $$ mathcal{U}f={(x,y)inmathbb{R}times [0,infty):0leq y<f(x)}. $$ The
function $f$ is Lebesgue measurable if $mathcal{U}f$ is Lebesgue measurable
with respect to the planar Lebesgue measure and then we define $$
int_{mathbb{R}} f=m_2(mathcal{U}f). $$

I find this approach quite nice if you want to have a quick introduction to the Lebesgue integration. For example:

You get the monotone convergence theorem for free: it is a
straightforward consequence of the fact that the measure of the union
of an increasing sequence of sets is the limit of measures.

As pointed out by Nik Weaver, the equality $int(f+g)=int f+int g$ is not obvious, but it can be proved quickly with the following trick:
$$
T_f:(x,y)mapsto (x,f(x)+y)
$$
maps the set $mathcal{U}g$ to a set disjoint from $mathcal{U}f$,
$$
mathcal{U}(f+g)=mathcal{U}f sqcup T_f(mathcal{Ug})
$$
and then

$$ int_{mathbb{R}} f+g= int_{mathbb{R}} f +int_{mathbb{R}} g $$

follows immediately once you prove that the sets $mathcal{U}(g)$ and
$T_f(mathcal{U}g)$ have the same measure. Pugh proves it on one page.

If $f: mathbb{R} to [0,infty)$ is Borel (or Lebesgue) measurable, then for each rational $a > 0$ define $X_a = f^{-1}([a,infty)) times [0,a)$. Then each $X_a$ is measurable and their union is exactly the region under the graph. So the region under the graph is measurable.

I think the reason why we develop the Lebesgue integral in the usual way is because it provides a powerful technique (characteristic functions --> simple functions --> arbitrary measurable functions) for deriving the basic theory of the integral. Even simple things like $int f + int g = int (f + g)$ aren't obvious if you take "measure of the region under the graph" as the definition.

The "area under a graph" approach is used in Wheeden/Zygmund's 1977 text Measure and Integral. An Introduction to Real Analysis, a book that was used (among other possible places) in the early 1980s for a 2-semester graduate real analysis course at Indiana University (Bloomington).

(second sentence of Chapter 5, on p. 64) The approach we have chosen [for the integral of a nonnegative function $f:E rightarrow [0, +infty],$ where $E subseteq {mathbb R}^n$ is measurable] is based on the notion that the integral of a nonnegative $f$ should represent the volume of the region under the graph of $f.$

I looked in the preface and elsewhere for any historical or literature citations about this approach and did not see anything relevant. Also, later in this book abstract measure and integration theory is developed in one of the standard ways.

Suppose your main interest is in constructing the Lebesgue integral over a general abstract measure space $(X,mu)$. From the usual definitions via simple functions, this is fairly straightforward, and one can prove the standard theorems (dominated convergence, etc) without too much trouble. But if you want to use a definition as "area under the graph", you have to take a detour and construct Lebesgue measure on $mathbb{R}$, which is a fair amount of work.

Indeed, one can sort of see that having Lebesgue measure is overkill for this task, since the whole point of Lebesgue measure is to be able to define the length of very complicated subsets of $mathbb{R}$. But for finding the product measure of the region under the graph of a function $f : X to mathbb{R}$ via a Fubini-like approach, the only subsets of $mathbb{R}$ you really need to measure are intervals, of the form $[0,f(x)]$. We don't need any fancy measure theory to tell us the length of such sets.

Granted, in most measure theory courses you will eventually want to construct Lebesgue measure on $mathbb{R}$ anyway, since an abstract theory should have good motivating examples. But from a certain point of view, having to do it first could obscure the fundamentally simple definition of the integral.