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I am learning about Riemann-Stieltjes Integrals from Carother's Real Analysis. Specifically, I'm trying to learn Riesz Representation. However, prior to that I need to learn the following Theorem:

Let $\alpha\in BV([a,b])$ and $B(x) = V_{a}^{x}(\alpha).$ Then the following holds: \begin{align*} \Big|\int_{a}^{b}fd\alpha\Big| \leq \int_{a}^{b}| f|d\beta \leq ||f||_{\infty}V_{a}^{b}(\alpha) \end{align*}

I understand most of the proof but one part of it is not obvious to me. I don't understand how the author makes the jump from saying $f \in R_{\alpha}([a,b])$ $\implies$ $f \in R_{B}([a,b])$. I believe they are using the following theorem, but I could be wrong:

Let $\alpha,\beta \in BV([a,b])$. Then the following holds: \begin{align*} f \in R_{\alpha}\cap R_{\beta} \implies f \in R_{\alpha \pm\beta} \end{align*}

Any help is greatly appreciated! Thanks again.

Tomislav
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  • Are you confused about how to prove that $f \in \mathcal{R}_B([a,b])$ or how to obtain the last inequality? I cover both below. The first part is not conceptually difficult but it involves a lot of technical details and estimates (see my linked answer). The proof of the inequality is where you use that integrability with respect to $\alpha, \beta$ implies integrability with respect to $\alpha \pm \beta$. – RRL Mar 10 '19 at 00:21
  • @RRL I was confused about the first part that you have provided in the link. Really appreciate that. The response you gave below was very helpful. – Tomislav Mar 10 '19 at 01:30

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With the function $B$ defined using the total variation of $\alpha$ on the interval $[a,x]$:

$$B(x) = V_a^x(\alpha),$$

it can be shown that $f$ is Riemann-Stieltjes integrable with respect to $B$ and $B - \alpha$. Also if $f$ is integrable then so is $|f|$. A proof is given here.

It appears you understand this fact, and your idea about how to proceed is correct.

To obtain the inequality, take $g = (B+ \alpha)/2$ and $h = (B-\alpha)/2$. Both are increasing functions on $[a,b]$ and using basic properties of Riemann-Stieltjes integration we have

$$\begin{align}\left|\int_a^b f \, d\alpha \right| &= \left|\int_a^b f \, d(g - h) \right| \\ &= \left|\int_a^b f \, dg - \int_a^b f \, dh \right| \\ &\leqslant \left|\int_a^b f \, dg \right| + \left|\int_a^b f \, dh \right| \\ &\leqslant \int_a^b |f| \, dg + \int_a^b |f| \, dh \\ & = \int_a^b |f| \, d(g + h) \\ &= \int_a^b |f| \, dB \\ &\leqslant \sup_{t \in [a,b]}|f(t)|(B(b) - B(a)) \\ &= \|f\|_{\infty}V_a^b(\alpha)\end{align}$$

RRL
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