What's the difference between the Riemann & Darboux Integrals?

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Discussion Overview

The discussion centers on the differences and similarities between the Riemann and Darboux integrals, exploring their definitions, equivalence, and implications in integration theory. The scope includes theoretical aspects of integration and mathematical reasoning.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants suggest that Riemann and Darboux integrals appear similar, questioning why they have different names if they are essentially the same.
  • Others argue that while the definitions of Riemann and Darboux integrals differ, they are equivalent in terms of the functions they can integrate.
  • A participant notes that Darboux integrals involve upper and lower sums, while Riemann integrals are based on a mean value approach.
  • One participant references a detailed explanation by Frederic Riesz regarding the types of Darboux integrals and their relationship to Riemann integrals, mentioning that the existence of both types of Darboux integrals can indicate integrability in the Riemann sense.
  • It is proposed that Darboux integrals can encompass a broader class of functions compared to Riemann integrals, depending on the existence of upper and lower integrals.

Areas of Agreement / Disagreement

Participants express differing views on the equivalence of Riemann and Darboux integrals, with some asserting their equivalence while others highlight their distinct definitions. The discussion remains unresolved regarding the implications of these differences.

Contextual Notes

The discussion does not clarify the specific conditions under which the equivalence of the integrals holds or the limitations of the definitions provided. There is also a lack of consensus on the broader implications of the differences in integrability.

swampwiz
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I was reading about this, and they seem the same. Of course, if they were the same, they wouldn't have different names.
 
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Riemann integrals and Darboux integrals have different definitions. However they are equivalent.
 
One of the two authors, Frederic Riesz, describes the question in detail in points 10 and 13 of Chapter I of the book by F. Riesz and B. Sz. Nagy, Leçons d'analyse functionelle.

There are two types of Darboux-integrals: upper and lower. These integrals belong to two interval functions

$$(\beta-\alpha) \sup_{\alpha \leq x \leq \beta} f(x), ~~~~~ (\beta-\alpha) \inf_{\alpha \leq x \leq \beta} f(x). $$
If the first interval function can be integrated, then we are talking about the upper Darboux-integral, if the second, then the lower Darboux-integral. Maybe both exist, perhaps just one, possibly none.

In cases where both exist and are even equal, it is said that the function can be integrated in the Riemann sense. So Darboux's integrals have much more functions than Riemann's integrals.
 

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