Riemann-Stieltjes Integrable

No. Take $\alpha(x) = 1/x^3$ and $f(x) = g(x) = x^2$. we have $$ \int_1^\infty f(x) d\alpha(x) = \int_1^\infty g(x) d\alpha(x) = -3\int_1^\infty \frac 1 {x^2} dx = -3 $$ but $$ \int_1^\infty f(g(x)) d\alpha(x) = -3\int_1^\infty dx = -\infty $$