# References on the equivalence of different definitions of integrability

While writing a chapter of a book about mathematical analysis, I decided to compare some definitions of integrability that are usually taught to sophomore students, in Italy. I briefly collect four popular definitions.

1. A bounded function $f \colon [a,b] \to \mathbb{R}$ is integrable if there exists a unique real number $I$ such that $I$ is larger than every Darboux lower sum for $f$ and smaller than every Darboux upper sum for $f$, i.e. if $$\underline{\int}_a^b f = \overline{\int}_a^b f.$$
2. A function $f \colon [a,b] \to \mathbb{R}$ is integrable if there exists a number $I$ such that, for every $\epsilon>0$ there is $\delta >0$ with the property that, for each partition $\mathcal{P}=\{x_k \mid k=0,\ldots, n\}$ of $[a,b]$ of size $|\mathcal{P}| < \delta$, any (i.e. for any choice of the nodes $t_k \in [x_{k},x_{x+1})$) Cauchy sum $\Sigma(\mathcal{P})=\sum_{k=0}^n f(t_k)(x_{k+1}-x_{k})$ satisfies $$\left| \Sigma(\mathcal{P}) - I \right| < \epsilon.$$
3. Same as 1., but considering only uniformly spaced partitions.
4. Same as 2., but considering only uniformly spaced partitions.

When I was a student myself, I was invited to believe that 2. was stronger than 1. (functions which are integrable according to 2. are less than those which are integrable with respect to 1.), and moreover that uniformly spaced partitions provide a proper subset of integrable functions. Some years later, I learned from these notes that this was false: the four definitions produce the same class of integrable functions.

I've been looking for a precise reference in the literature, possibily in english. I think this should be written in some book, but I could not find any, at least among those I can find in my library. I would be glad to read suggestions and bibliographical refereces about this elementary topic.

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How do you define the size of a partition? Largest radius of a set in the partition? –  Alex Becker Jul 4 '12 at 8:09
Yes, I did not want to write a very long question, and I omitted some details: $$|\mathcal{P}| = \max_{k} (x_{k+1}-x_{k}).$$ –  Siminore Jul 4 '12 at 8:10