1
$\begingroup$

While doing research on the construction of the Riemann integral I've stumbled upon two different criterions for Riemann integrability. In this article (on page $19$) there is a theorem called the Cauchy Criterion for Henstock Integrals. This theorem states that $f:[a,b]\rightarrow \mathbb{R}$ is Henstock integrable on $[a,b]$ if and only if $\forall \varepsilon > 0 \exists\delta:[a,b]\rightarrow \mathbb{R^+}$ such that when $P_1$ and $P_2$ are any $\delta$-,fine tagged partitions of $[a,b]$ then $|S(f,P_1,t_1)-S(f,P_2,t_2)|<\varepsilon$, where $t_1$ and $t_2$ are tags. This theorem evidently applies for Riemann integrals as well if we let $\delta$ be constant. So with this slight tweak I suppose we could call the theorem the Cauchy criterion for Riemann integrability.

However, at other places on the internet I've seen this theorem be stated as $\forall \varepsilon>0$ there exists partitions $P_1$ and $P_2$ of $[a,b]$ such that $U(f,P_1)-L(f,P_2)<\varepsilon$.

I am aware that both the aforementioned theorems are equivalent. In terms of how these two theorems are stated I don't think it makes sense for them to be called the same. The second theorem should probably be called the Cauchy criterion for Darkboux integrability instead. However, I feel like it's a bit arrogant of me, as an undergraduate student of mathematics to name theorems that are not due to my own work. So if I published a paper on integration, and named the second theorem after Cauchy and Darboux, do you think anyone would object?

$\endgroup$
1
$\begingroup$

The theorem you refer to is called Cauchy's condition for Riemann integrability and is more on the lines of Cauchy's general principle of convergence.

The theorem related to difference between upper and lower Darboux sums is called Riemann's condition for Riemann integrability and it was explicitly mentioned by Riemann while developing a theory of his integral.

These two theorems are different in the sense that one deals with Riemann sums (the Cauchy's condition) and the other deals with Darboux sums.

Regarding your question about publishing I have no idea as I haven't published any paper. But I think if one explicitly mentions the terms used (including names of theorems) this would remove any ambiguity and should not be objected.

$\endgroup$
  • $\begingroup$ I didn't know that the second theorem was due to Riemann. Thank you very much. $\endgroup$ – David Oct 29 '18 at 8:13
  • 1
    $\begingroup$ @David: You should have a look at Bressoud's A Radical Approach to Lebesgue's Theory of Integration for many historical remarks on the development of Riemann and Lebesgue integrals. $\endgroup$ – Paramanand Singh Oct 29 '18 at 8:25
  • 1
    $\begingroup$ @David: Riemann also did a deeper analysis of difference between upper and lower Darboux sums to prove a theorem which is almost equal to the condition of Riemann integrability based on sets of measure zero. $\endgroup$ – Paramanand Singh Oct 29 '18 at 8:27
  • $\begingroup$ How Riemann though of all these things given the time he lived in just blows my mind. Thank you for the recommendation, I will have a look at it. $\endgroup$ – David Oct 30 '18 at 5:39

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.