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!

  • 3
    $\begingroup$ All the upper sums $U(f,P)$ are larger than the intended integral. So to get something that should be called the integral, take their infimum. Work out some examples. Say $f(x) = x$ on $[0,1]$. $\endgroup$ – GEdgar May 30 '15 at 20:30

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.


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