There are "plenty" of definition of Riemann integrability for a real function. For example:
$\forall \epsilon>0,~\exists \delta >0$, $\forall P$: partition of $[a,b]$, $\forall T$: sample points of $P$, ($\lVert P\rVert<\delta\Rightarrow |S(f,P,T)-I|<\epsilon$).
$\forall \epsilon>0,~\exists P$: partition of $[a,b]$, $\forall P'$: refinement of $P$, $\forall T$: sample points of $P'$, $|S(f,P,T)-I|<\epsilon$.
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As Wikipedia said, the first criterion is "very difficult to use". However, I've seen many books didn't even mention the second criterion, just stating the first with Darboux integral(i.e. $U(f,P)-L(f,P)<\epsilon$ or "upper integral=lower integral"), or stating the first with Cauchy-criterion(i.e., when the mesh is small enough, the two Riemann Sum are very closed to each other). Why does wiki say so? Concrete examples that can point out why the second is more convenient are welcomed.
And on the other hand, I want to ask that, don't you think the second criterion is not very "free" and natural? Since it just guarantee the Riemann sum of all refinement of some fixed partition $P$ is closed to $I$, but if you have a partition that is not related to $P$, even its mesh is very very small, you can't conclude that its Riemann sum is closed to $I$. So doesn't the first criterion have better intution?