My book defines the upper and lower Darboux sums $U(f,P)$ and $L(f,P)$ respectively then follows up with a confusing definition of the upper and lower Darboux integrals $U(f)$ and $L(f)$ respectively. Before I ask my question I should define some notation:
Here we partition the interval $[a,b]$ by the subset $P$ having the form $P = \{ a=t_0<t_1<\cdots <t_n = b\}$. Also, $M(f,S) = \sup\{ f(x):x\in S\}$ and $m(f,S)=\inf\{ f(x):x\in S\}$.
The definition of the upper Darboux sum is :
- $U(f,P) = \sum_{k=1}^{n}M(f,[t_{k-1}, t_k])\ (t_k-t_{k-1})$
The definition of the lower Darboux sum is :
- $L(f,P) = \sum_{k=1}^{n}m(f,[t_{k-1}, t_k])\ (t_k-t_{k-1})$
Now, here comes my question. When the book defines the upper Darboux integral it says:
$U(f) = \inf\{ U(f,P) : P$ is a partition of $[a,b]\}$.
Why does it switch to say inf instead of sup like in the previous definition of the sum? How can I conceptually/intuitively understand the difference between the sum and integral? Do you guys have any other literature I could read up on to make me feel more comfortable about this topic?
Thank you!
Infimum of the set of upper sums decreses as you add partitions and you want the smallest upper "approximation". Further the integral is a real number which is obtained as inf/sup of a sets of sums, it is therefore is is not a sum.