Real Analysis: I need help understanding the textbook's definition of upper and lower integrals

Question

Can you explain how the lower integral is the supremum of a sum? Isn't $L(f,P)$ just a sum and therefore a single number? Definition 5.13 says from $a$ to $b$, not from each $x_j$ to $x_{j—1}$. And it says for a given partition, not finer and finer partitions.

I'm just very confused on these definitions and would love if someone could rephrase it in a way that matches this definition. (I know there's another way to define it in terms of limits but I need to understand it the way this textbook defines it.)

Textbook definition of Riemann sums

Textbook definition of upper and lower integrals

Answer 1

Your textbook's definition of upper and lower integrals says "over all partitions". That's what the bit on the end means.

Basically, "try out all partitions and take the lowest value you get or approach" for the upper integral.

Answer 2

Yes, $L(f,P)$ is a single number for a particular partition $P$, but the notation $$\int_{a}^{b}f(x)\,dx = \sup\left\lbrace L(f,P): P \text{ is a partition of }[a,b]\right\rbrace $$ means the supremum over all partitions of $[a,b]$. In plain(er) English, $\sup\left\lbrace L(f,P): P \text{ is a partition of }[a,b]\right\rbrace$ means "the supremum of the set of numbers $L(f,P)$ for every partition $P$ of $[a,b]$."