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?