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Let $\{\alpha_i\}$ be a sequence of monotonically increasing functions on $[a,b]$.

Suppose $f\in \mathscr{R}(\alpha_i), \forall i\in \mathbb{N}$ and $\sum \alpha_i$ is convergent.

Let $A=\sum_{i=1}^{\infty} \alpha_i$

Then $f\in \mathscr{R}(A)$?

If it is not true, then what if $f$ is cotinuous at every discontinuity of each $\alpha_i$?

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Could you precise what you mean by $\mathscr R (\alpha_i)$? – Sebastien B Nov 28 '12 at 10:07
    
@Sebastien "Riemann-Stieltjes Integrable on $[a,b]$ with respect to $\alpha$" – Katlus Nov 28 '12 at 10:08

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