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

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

• The supremum is taken over all possible partitions for the lower integral. Likewise for the upper integral, but with infimum. May 11 '21 at 18:13
• For fixed $f$, $L(f,P)$ is still a function of the chosen partition $P$. So, it is a sum involving $f$ being evaluated at points of the partition $P$. In the definition of a lower integral, you take a supremum over all partitions $P$. May 11 '21 at 18:14

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 $$[a,b]$$."