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At some point, wikipedia says that

An equivalent and sometimes useful criterion for the integrability of $f$ is to show that for every $ε > 0$ there exists a partition $P_ε$ on $[a,b]$ such that

$$U_{f,P_{\epsilon}}-L_{f,P_{\epsilon}} < \epsilon.$$

My question has to do with using $\epsilon$ both for the upper and lower sums and also for the partition (Intuitively, the physical units would not match...).

Shouldn't we use instead a epson-delta definition like the following?

$$\forall \epsilon>0\quad\exists\delta: U_{f,P_{\delta}}-L_{f,P_{\delta}} < \epsilon.$$

Are they equivalent?

Miguel
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  • They are just numbers (they aren't free men), no units in sight. (Besides, $\varepsilon$ is just an index of $P_{\varepsilon}$.) – Daniel Fischer Jan 29 '16 at 19:30
  • @Daniel I was understanding $P_\epsilon$ as meaning the partition where the max subinterval has size $\epsilon$. Not sure what it means. – Miguel Jan 29 '16 at 19:43
  • No, it's just some partition for which the upper and lower Darboux sum differ by less than $\varepsilon$. If $f$ is constant on long subintervals, the mesh of the partition may be quite large. – Daniel Fischer Jan 29 '16 at 19:46
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    They wrote $P_\epsilon$ meaning some partition (depending on $\epsilon$.) We do not know, or need to know, anything about the mesh of that partition. – GEdgar Jan 29 '16 at 20:09

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