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Theorem 2.24 of Measure and Integral by Wheeden and Zygmund states that if $f$ is continuous on $[a,b]$ and $\phi$ is of bounded variation on $[a,b]$, then $\int_a^b f d\phi$ exists. Moreover, $$\left\vert \int_a^b f\,d\phi \right\vert \le \left(\sup_{[a,b]} \vert f\vert \right) V[\phi;a,b].$$

A detailed proof for the first part is provided. We have the relationship $L_\Gamma \le R_\Gamma \le U_\Gamma$, hence we can show that $\lim_{|\Gamma| \to 0} L_\Gamma$ and $\lim_{|\Gamma| \to 0} U_\Gamma$ exist and are equal.

For the second part, $\vert \int_a^b f\, d\phi\vert \leq (\sup_{(a,b)} \vert f\vert) V[\phi;a,b]$, they say it follows from a similar one for $R_\Gamma$ by taking the limit. What does it mean? How to show the second part?

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user398843
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    More directly, $\left|\int_a^b f,d\phi\right|\le \int_a^b |f|,d\phi \le \sup|f| \int_a^b d\phi$. – Ted Shifrin Jul 08 '18 at 21:13
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    To the OP, what is $V[\phi;a,b]$ when $\phi$ is monotone increasing? (This is assumed in the first line of the proof.) –  Jul 08 '18 at 21:30
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    Another proof that $\vert \int_a^b f, d\phi\vert \leq (\sup_{(a,b)} \vert f\vert) V[\phi;a,b]$ when $\phi$ is BV but not monotone is given here – RRL Jul 08 '18 at 21:34

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