What does the lower integral mean?

Question

Defintion: ${}_{-}\int_a^b f(x) \,\text{d}x = \sup \{L(f,p) \mid \text{$p$ is a partition of $[a,b]$} \}$ where $L(f,p)$ is the lower sum.

My problem: I am confused with the definition itself of the lower integral as $L(f,p)$ should be a number, so how would I take the supremum of a number?

Answer 1

For each partition $p$, $L(f, p)$ is a (real) number. For different partitions $p$, you get (potentially) different results for $L(f, p)$. $$ \{L(f, p)\mid p \text{ is a partition of }[a, b]\} $$ is the set of all those possible reults. You are supposed to take the supremum of this set of numbers.

Answer 2

$L(f,p)$ is a number, sure, but

$$\{L(f,p)|p\text{ is a partition of }[a,b]\}$$ is a set of numbers, and the definition takes the supremum of this set.