I am re-reading lecture notes and I am having a few issues. Okay, I understand the concept of a partition. You basically have this bounded function $f:[a,b] \rightarrow \mathbb{R}$ and we are creating the idea of the integral out of scratch in order to do calculus with it. Then he introduced the concept of a lower and upper darboux sum, where you split your partition into partition intervals divided between partition points. So a Lower Darboux Sum, for example, is the summation of the infimums of each partition interval multiplied by the length of the partition interval.
However, the following page it begins to talk about "lower and upper integrals" and one of the very first explanations here I am having trouble understanding what it "looks like."
It says the lower integral of f from a to b is equal to the supremum{LowerSum(f,Partition) | P a partition of [a,b]}. What does this mean at all?
The way I understand it right now is basically: http://www.math.hmc.edu/calculus/tutorials/riemann_sums/gif/figure6.gif If you were to add all the blue squares together, that's your "Lower Darboux Sum" but what is the jump to having the sup of this "Lower Sum". To me, the Lower Sum is just a number, so what is the meaning behind supremum-ing it?
Also, there was a proof that I was supposed to do that said verbatim "Supposed that the bounded function f:[a,b] to R has the property that $f(x) \geq 0$ for all x \in [a,b]. Prove that the lower integral of f is always greater than or equal to zero. Geometrically, it seems obvious, and I basically just explained that the LowerSum is equal to the summation of each individual infimum multiplied by the difference in length of the interval in question, but I was wondering if there was some way of more clearly (at least to me) stating that "well, it seems geometrically equivalent, and because your function is always positive, it implies that this area you are calculating will also be positive."
Here is a photo of my textbook with what I'm talking about in the beginning of this post, the proof is not as important presently: http://oi49.tinypic.com/e8olt3.jpg
I appreciate any insight.