A Riemann Stieltjes integral with absolute value

Question

Suppose that f is Riemann-integrable on $[-5,5]$ showw that $f$ is Riemann-Stieltjes integrable over $g(x)=|x|$ and $$\int_{-5}^{5}fdg=\int_{0}^{5}f(x)dx-\int_{-5}^{0}f(x)dx$$

please any help would be usefull.

Answer 1

Start with $\int_{-5}^5 f dg = \int_{-5}^0 fdg + \int_0^5 f dg $.

Note that $g(x) = x$ for $x \ge 0$ so $\int_0^5 f dg = \int_0^5 f(x) dx$, the usual Riemann integral.

For $x \ge 0$, we have $g(x) = -x$, so working from the definition we see that $\int_{-5}^0 fdg = - \int_{-5}^0 f(x) dx$.

I am a little lax above, in that I should establish the existence first, but this is straightforward working from the definition and given the specific form of $g$ on $x \le 0$ and $x \ge 0$.