0
$\begingroup$

If $f(x) = x- \left \lfloor{x}\right \rfloor $ and $\alpha(x) = x^2$, show that f is Riemann-Stieltjes Integrable on [-1,2].

Every proof in my textbook seems to show that $\alpha$ is increasing on its domain and then shows f is RS-I, but that does not seem to be the case here. Since $\alpha$ is not increasing, how do I go about proving this?

$\endgroup$
0
$\begingroup$

$\alpha$ increasing is a very usual case, but being of bounded variation (difference between two monotone functions) is enough. Writing $\alpha$ as difference between two monotone functions is very easy in this case.

As $\alpha(x) = x^2$ is $C^1$, another property is applicable: $$\int_a^b fd\alpha = \int_a^b f\alpha'$$ where the RHS is simply a Riemann integral. In this case, the RHS exists because $f$ has a finite mumber of discontinuities.

See Riemann–Stieltjes integral

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.