I am starting to study some Measurement Theory following the book Measures, Integrals and Martingales and I have a doubt regarding the definition of integral that the author provides.
First he defines the integral for simple functions as
$$I_{\mu}(f):=\sum_{j=0}^M y_j \mu(A_j)$$
(where $f$ is a simple function, ${A_j}\in \sigma$-algebra and $\mu$ is a measure).
Then using the previous one he defines (Definition $9.4$) the $\mu$-integral of a positive measurable function as:
$$\int u \, d\mu:=\sup\{I_{\mu}(g):g\leq u, g \text{ is simple}\}$$
I am not really familiar with this definition, but this looks pretty much as the "lower integral" on Riemann's definition.
Actually I was expecting to see also a sort of "upper integral" too, but nothing else was stated.
Does this have to do with the fact that the function $u$ is measurable (by assumption)?
If not, why don't we require both the upper and the lower integral to coincide?
Maybe this is a straightforward application of some well-known result that I am not aware of, but as I said this is my first approach to measure theory.
Thanks in advance
EDIT: Could this be because we know that any measurable function is
the point-wise limit of simple functions, so we don't actually need to
calculate the upper integral. We can ‘inscribe’ simple functions
below the graph of $u$ and exhaust the area?
Best Answer
Yes, the idea is to measure the area below the graph using simple functions which approximate the function from below. There is a general staatement which shows that any measurable function $u \geq 0$ can be approximated from below by simple functions (in the book you are using, this result is called Sombrero lemma), and therefore this is a reasonable approach.
If you, say, have a bounded function $u \geq 0$, then you can use the same idea as in the proof of the Sombrero lemma to construct a sequence of (uniformly bounded) simple functions $g_j$, $j \geq 1$, such that $g_j \downarrow u$ and the integrals will converge, i.e. $\int g_j \, d\mu \to \int u \, d\mu$. This shows that the "upper integral" and "lower integral" coincide in this case.
For unbounded functions it is typically impossible to get an approximation from above by simple functions; that's, in fact, one of the reasons to define the integral as an "lower integral". The point is that any real-valued simple function $g$ is bounded. Therefore, we can in general not find simple functions $g$ with $g \geq u$ for $u$ unbounded. In principle, we might allow $g$ to take the value $+\infty$ but if this happens on a set of positive measure, then the integral $\int g \, d\mu$ will be infinite. If we would use upper integrals to define the integral $\int u \, d\mu$, then this would mean that only bounded functions were integrable - and that's hardly what we want to have.