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I've been working through a book on Introductory Real Analysis and I've been stumped by part of this problem.

Suppose we consider a partition that splits $[a,b]$ into $n$ partitions each with length $\frac{b-a}{n}$. Show that a function $f:[a,b]\rightarrow\mathbb R$ is Darboux integrable on just these partitions iff it is Darboux integrable over all partitions.

The if direction is very trivial, that one isn't too tricky.

However, the only if direction has me stumped. How can I show that integrability over this type of partition implies integrability over any partitions. I think I can show that it gives integrability over rational partitions (not sure if that's a formal term, basically a partition where all interval lengths are rational). However, I'm having trouble extending it to all partitions in general.

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    $\begingroup$ The supremum over a set is always greater than or equal to the supremum over a subset, etc. $\endgroup$
    – RRL
    Nov 7, 2020 at 7:41

1 Answer 1

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Let $\mathcal{P}$ ($\mathcal{P}_U$) be the collection of all partitions (uniform partitions) of $[a,b]$.

For a bounded function $f$, we have $L(P,f) \leqslant U(Q,f)$ for lower and upper Darboux sums corresponding to arbitrary partitions $P$ and $Q$. It follows that

$$\sup_{P \in \mathcal{P}}L(P,f) \leqslant \inf_{P \in \mathcal{P}}U(P,f), \quad \sup_{P \in \mathcal{P}_U}L(P,f) \leqslant \inf_{P \in \mathcal{P}_U}U(P,f)$$

Since $\mathcal{P}_U \subset \mathcal{P}$ we have

$$\{U(P,f) \,|\, P \in \mathcal{P}_U\} \subset \{U(P,f) \,|\, P \in \mathcal{P}\}, \quad \{L(P,f) \,|\, P \in \mathcal{P}_U\} \subset \{L(P,f) \,|\, P \in \mathcal{P}\}$$

Hence,

$$\tag{*}\sup_{P \in \mathcal{P}_U}L(P,f) \leqslant \sup_{P \in \mathcal{P}}L(P,f) \leqslant \inf_{P \in \mathcal{P}}U(P,f) \leqslant \inf_{P \in \mathcal{P}_U}U(P,f)$$

Darboux integrability with respect to uniform partitions means that

$$\sup_{P \in \mathcal{P}_U}L(P,f) =\inf_{P \in \mathcal{P}_U}U(P,f),$$

which, in view of (*), implies that $f$ is Darboux integrable with respect to all partitions since

$$0 \leqslant \inf_{P \in \mathcal{P}}U(P,f)- \sup_{P \in \mathcal{P}}L(P,f) \leqslant \inf_{P \in \mathcal{P}_U}U(P,f)- \sup_{P \in \mathcal{P}_U}L(P,f) = 0,$$ and $$ \inf_{P \in \mathcal{P}}U(P,f)= \sup_{P \in \mathcal{P}}L(P,f) $$

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  • $\begingroup$ Ok, this makes sense, but I think I made a big error in my proof of the opposite direction then. Would you be able to give a hint for that direction? $\endgroup$
    – Alexander
    Nov 7, 2020 at 8:26
  • $\begingroup$ @AyushTibrewal: Yes -- that direction is not trivial. First, there is an equivalent condition for Darboux integrability. We have $f$ Darboux integrable if for any $\epsilon>0$ there is a partition $P_\epsilon$ such that $U(P_\epsilon,f) - L(P_\epsilon,f) < \epsilon$. So assuming that the Darboux integral exists with respect to all partitions we want to show that there is always a uniform partition $P_U$ such that $U(P_U,f) - L(P_U,f) < \epsilon$. $\endgroup$
    – RRL
    Nov 7, 2020 at 8:53
  • $\begingroup$ One way to proceed is to show that an equivalent condition is for any $\epsilon > 0$ there exists $\delta > 0$ such that for all partitions $P$ with mesh (length of longest subinterval) less than $\delta$ we have $U(P,f) - L(P,f)< \epsilon$. It is not an easy proof. I indicated how to do it here. $\endgroup$
    – RRL
    Nov 7, 2020 at 8:57
  • $\begingroup$ With that result, you can always find a uniform partition with mesh less than $\delta$ by taking $n$ sufficiently large. Let me think if there is an easier way. $\endgroup$
    – RRL
    Nov 7, 2020 at 8:59
  • $\begingroup$ Ok, I think I got a proof for the if direction through dyadic partitions since dyadics are a subset of the uniform partitions. I think that works since I've already proven that dyadic integrability is equivalent to Riemann and thus Darboux. From there, its similar where if a subset of the uniform partitions (dyadics of course) have defined integrability, so does the larger set of all uniform partitions. $\endgroup$
    – Alexander
    Nov 7, 2020 at 18:50

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