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In my book it has been proved that the existence of the limit $$\lim_{||P|| \rightarrow 0} S(P,f,\alpha)$$ is sufficient for RS integrability of $f$ w.r.t. $\alpha$ on $[a,b]$.But is it necessary condition for RS integrability?I have a confusion regarding this.Please help me.

Thank you in advance.

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    How do you define RS integrability? Many books define RS integrability using the limit in your question and in that case your question is trivial. Note further that unlike the case Riemann integration we don't have an easy to use necessary and sufficient criterion of RS integrability. – 2017-02-15

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