Prove if a bounded function is integrable the difference between the upper sum and lower sum of the regular partition tends to 0.

Question

How do I prove that a condition for (Riemann) integrability is that the difference between the upper and lower sums of the regular partition $D_n:0=a_0<a_1<...<a_n=1$ where $a_k=(k/n)$ tends to 0 as n tends to infinity? I've tried used the Riemann criterion, so for all $ε>0$ there is a dissection $D$ such that the difference between the upper and lower sums of $D$ is less than $ε$, and this is fine in the case where $D$ is composed only of rationals, but I can't figure out how to make the argument rigorous in the case that $D$ may contain irrationals. Any help would be greatly appreciated! :)

Answer 1

First, it is a sufficient condition.

As long as $f$ is bounded on $[0,1]$, the upper and lower sums corresponding to arbitrary partitions are bounded. Let $\mathcal{P}$ denote the set of all partitions of $[0,1].$ Consequently, the sets $\{L(P,f): P \in \mathcal{P}\}$ and $\{U(P,f): P \in \mathcal{P}\}$ are bounded, and this guarantees the existence of

$$\underline{\int}_0^1 f(x) \, dx = \sup_{P \in \mathcal{P}}\, L(P,f), \\ \overline{\int}_0^1 f(x) \, dx = \inf_{P \in \mathcal{P}}\, U(P,f) , $$

which are called the lower and upper integrals.

Given any regular partition $D_n$ we have

$$L(D_n,f) \leqslant \underline{\int}_0^1 f(x) \, dx \leqslant \overline{\int}_0^1 f(x) \, dx \leqslant U(D_n,f).$$

The central inequality follows because for any partitions $P$ and $Q$ we have $L(P,f) \leqslant U(Q,f)$ (take a common refinement of the partitions to show this) and, thus $\sup_{P \in \mathcal{P}} \,L(P,f) \leqslant \inf_{Q \in \mathcal{P}} \,U(Q,f)$.

Hence,

$$0 \leqslant \overline{\int}_0^1 f(x) \, dx - \underline{\int}_0^1 f(x) \, dx \leqslant U(D_n,f) - L(D_n,f).$$

The right-hand side converges to $0$ as $n \to \infty$, by hypothesis, which implies that $f$ is integrable since we must have

$$\underline{\int}_0^1 f(x) \, dx = \overline{\int}_0^1 f(x) \, dx, $$

where the common value of lower and upper integrals is by definition the value of the integral.

To show it is a necessary condition, consider

$$\left|U(D_n,f) - L(D_n,f) \right| \leqslant \left|U(D_n,f) - \int_0^1 f(x) \, dx \right| + \left|L(D_n,f) - \int_0^1 f(x) \, dx \right|.$$

The two terms on the RHS go to zero as $n \to \infty$. This is a consequence of the equivalent condition for integrability where for arbitrary Riemann sums corresponding to tagged partitions we have

$$\tag{*}\int_0^1 f(x) \, dx = \lim_{\|P\| \to 0} S(P,f).$$

Here $\|P\| = \max_{1 \leqslant j \leqslant n} (x_j - x_{j-1})$ is the norm of the partition $P = (x_0,x_1, \ldots, x_n)$ and, clearly, $\|D_n\| \to 0$ if and only if $n \to \infty$.

It takes a bit of effort to prove the equivalence of $(*)$ to the definition of the Riemann integral in terms of partition refinement or the Darboux approach. It has been shown a number of times on this site including here.