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Here is my question.

Let $\alpha$ be a monotonically increasing function on $[a,b]$ and $f$ be a real function defined on $[a,b]$.

Suppose;

(1) $\forall t\in (a,b), f\in\mathscr{R}(\alpha)$ on $[t,b]$.

(2) $\lim_{t\to a} \int_{t}^{b} f d\alpha$ is finite

Then, I want to prove $f\in\mathscr{R}(\alpha)$ on $[a,b]$.

It seems i need to prove boundedness of $f$ on $[a,b]$ first, but i don't know how to approach this problem. Please show me how to prove this or give me a hint.. Thank you in advance!

EDIT;

It seems argument for this is exactly the same as that when $\alpha$ is an identity map. So, you could assume $\alpha$ to be an identity map too!

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Let $\alpha(x) =x$, $f(x) = \frac{1}{\sqrt{x}}$ (except at $x=0$ where we define $f(0) = 0$). Then $f$ satisfies (1), (2) above with $a=0$, $b=1$, but $f$ is unbounded near $0$, hence $f\notin\mathscr{R}(\alpha)$ on $[0,1]$. –  copper.hat Dec 24 '12 at 3:53
    
@copper.hat Thank you! –  Katlus Dec 24 '12 at 5:07
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