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Suppose that $f$ and $g$ are both real-valued functions of bounded variation on $[-\infty,\infty]$. I have to estimate $$I :=\int_{-\infty}^\infty f\,\mathrm dg.$$ Do we have the estimate $$\vert I\vert \leq \Vert f\Vert_\infty\cdot\Vert g\Vert_\infty,$$ where $\Vert\cdot\Vert_\infty$ denotes the $\sup$-norm? Of course, this estimate is not useful unless $f$ or $g$ are bound, but I am willing to assume boundedness (as far as I know, bounded variation does not imply boundedness on all of $[-\infty,\infty]$).

I am not very experienced with functions of bounded variation, which makes it difficult to come up with a solution. My idea was: $$\vert I\vert = \left\vert\int_{-\infty}^\infty f\,\mathrm dg\right\vert \leq \int_{-\infty}^\infty\vert f\vert\,\mathrm dg \leq \int_{-\infty}^\infty \Vert f\Vert_\infty\,\mathrm dg = \Vert f\Vert_\infty g.$$ But this has to be false as the left hand side is a real number, while the right hand side is a function. Moreover, the right hand side can be negative, while the right can't (despite the inequality!). I suppose that the triangle inequality is the issue. But, perhaps, is the above estimate still correct and it's just my appraoch that is flawed.

Quertiopler
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  • What definition of bounded variation are you using? By my definition (see also the tag-info), bounded variation implies boundedness. To get the estimate, it may be helpful to first consider the case that $g$ is continuously differentiable, so that you have to estimate $$\biggl\lvert \int_{-\infty}^{+\infty} f(x)g'(x),dx\biggr\rvert,.$$ – Dermot Craddock Jun 19 '25 at 14:35
  • generally speaking $|\int_X f(x)d\mu(x)| \le ||f||_{\infty}|\mu|(X)$ where $|\mu|=\mu^++\mu^{-}$ is the total variation of $\mu=\mu^+-\mu^-$ written as the difference of two positive measures (one needs some finiteness condition eg at least one of $\mu^{+-}$ is finite or we need to take some limits in some sense - eg Cauchy) so as noted above it depends on your definition of bounded variation – Conrad Jun 19 '25 at 15:25
  • I see. I am aware of the inequality, but did not think about applying it to bv functions as integrators. But of course, these functions define a measure, which then is subject to this inequality. I think I will have to refine my conditions on $f$ and $g$ to obtain a bound. Unfortunately, a general bound does not seem to be obvious. Thank you! – Quertiopler Jun 19 '25 at 22:30

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In general, the estimate $$ \left| \int f \, dg \right| \leq \|f\|_\infty \cdot \|g\|_\infty $$ does not hold for the Riemann–Stieltjes integral, even when both $f$ and $g$ are bounded.

Let $N \in \mathbb{N}^\star$ and consider a function $f\colon \mathbb{R} \to [0,1]$ of class $C^1$, supported in $[0,2N]$, such that: $$ f(k) = \begin{cases} 0, & \text{if } k \in \{0,2,4,\dots,2N\}, \\ 1, & \text{if } k \in \{1,3,\dots,2N-1\}. \end{cases} $$

Now define a function $g\colon \mathbb{R} \to \mathbb{R}$ of bounded variation, also supported in $[0,2N]$, by $$ g(x) = \begin{cases} 0, & \text{if } x \in [2k,2k+1), \quad 0 \leq k < N, \\ 1, & \text{if } x \in [2k+1,2k+2), \quad 0 \leq k < N, \\ 0, & \text{otherwise}. \end{cases} $$

Both $f$ and $g$ are bounded, with $$ \|f\|_\infty = \|g\|_\infty = 1, $$ but the Riemann–Stieltjes integral satisfies $$ \int f \, dg = f(1) + f(3) + \dots + f(2N - 1) = N, $$ which clearly shows that the integral can exceed the product $\|f\|_\infty \cdot \|g\|_\infty$.

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Daniel Smania
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  • the comments already made me doubt. I think I get the message of the example, but there are two things that are unclear to me: i) what is $\mathbb N^*$? all integer including zero or including infinity? ii) I don't really get the definition of $g$. $k$ is running through all integers? – Quertiopler Jun 19 '25 at 22:28
  • @Quertiopler i. $N$ is a positive integer. ii. $k$ is running between $0$ and $N.$ In the example $N=7$ and and both functions are $0$ outside $[0,14]$ – Daniel Smania Jun 19 '25 at 23:18