# Sufficient condition for Riemann integrability

There are many ways to define the Riemann Integral. I am using this one, where I denote $$\sigma(f,P^{*})$$ the Riemann Sum relative to a tagged partition $$P^{*}$$:

$$\textbf{Definition}$$

We say that a function $$f:[a,b] \to \mathbb{R}$$ is Riemann-Integrable if exist the limit:

$$I=\lim_{||P|| \to 0} \sigma(f,P^{*})$$

and then we write $$I=\int_{a}^{b}f(x)dx$$.

The limit exist in the sense that given $$\epsilon > 0$$, there is $$\delta > 0$$ such that for any partition $$P$$ of $$[a,b]$$ with $$||P|| < \delta$$ and for any tagged partition $$P^{*}$$, we have:

$$|\sigma(f,P^{*}) - I| < \epsilon$$

By definition, if $$f$$ is integrable in $$[a.b]$$, then given $$\epsilon < 0$$ exists two tagged partition $$P^{*}$$ and $$P^{**}$$ such that:

$$|\sigma(f,P^{*}) - I| < \epsilon / 2$$

$$|\sigma(f,P^{**}) - I| < \epsilon / 2$$

hence, $$|\sigma(f,P^{*}) - \sigma(f,P^{**})| < \epsilon$$. Therefore, is a necessary condition for integrability that one can find a partition $$P$$ such that two Riemann Sums relative to $$P$$ are very close together, no matter what the scalars that we pick in $$P^{*}$$ and $$P^{**}$$.

I am pretty sure that this is a sufficient condition as well. But I have any ideas how to prove it. Can anyone help with this?

• It seems valid since one can choose a Riemann sum arbitrarily close to a Darboux sum. – Tony Piccolo Nov 24 '18 at 6:58

## 2 Answers

There is such a sufficient (Cauchy) condition. The correct statement is:

Suppose for any $$\epsilon > 0$$ there exists $$\delta > 0$$ such that $$|\sigma(f,P) - \sigma(f,P')| < \epsilon$$ for all partitions $$P$$ and $$P'$$ with $$\|P\|, \, \|P'\| < \delta$$ and for any choice of tags. Then $$f$$ is Riemann integrable.

In proving this one must first show the existence of a viable candidate for the value of the integral and then show that it satisfies the required definition.

First -- and I will leave this to you for now -- construct a decreasing sequence of positive numbers $$\delta_n$$ and partitions $$P_n$$ with $$\|P_n\| < \delta_n$$ such that for any partition $$P$$ with $$\|P\| < \delta_n$$ we have (for any choice of tags)

$$|\sigma(f,P) - \sigma(f,P_n)| < 1/n$$

Hence, if $$m \geqslant n$$ then $$|\sigma(f,P_m) - \sigma(f,P_n)| < 1/n$$. The sequence $$\sigma(f,P_n)$$ is a Cauchy sequence and must converge to a real number $$I$$.

To show that $$I$$ satisfies the definition of the integral $$\int_a^b f(x) \, dx$$, for any $$\epsilon >0$$ take $$n$$ such that $$1/n < \epsilon/2$$ and sufficiently large such that $$|\sigma(f,P_n) - I| < \epsilon/2$$. If $$P$$ is a partition with $$\|P\| < \delta_n$$, then it follows that

$$|\sigma(f,P) - I| \leqslant |\sigma(f,P) - \sigma(f,P_n)| + |\sigma(f,P_n) - I| < 1/n + \epsilon/2 < \epsilon$$

It is a modified version of the standard Cauchy criterion.

The crucial statement is

Let $$f:[a,b] \to \mathbb{R}$$ be a bounded function.
If for every $$\varepsilon>0$$ there exists a partition $$P$$ of $$[a,b]$$ such that $$|\sigma (f,P^*)- \sigma (f,P^{**})|<\varepsilon$$ for any Riemann sums $$\sigma (f,P^*)$$ and $$\sigma (f,P^{**})$$ for $$f$$ associated with $$P$$, then $$S(f,P)-s(f,P)<3\varepsilon$$.

It is about using the fact that $$S(f,P)$$ and $$s(f,P)$$ are respectively the supremum and the infimum of the set of Riemann sums for $$f$$ associated with $$P$$. The triangular inequality does the rest.