# Problem on Riemann-Stieltjes Integration and function approximation

Let $$\alpha$$ be continuous and increasing function on [a,b]. Given $$f$$ $$\in$$ $${R_\alpha }$$[a,b] and $$\epsilon$$> 0;

Then, Prove that there exist

(i) a step function $$h$$ on [a,b] with $${||h||_\infty}\leq{||f||_\infty}$$ such that $$\int\limits_a^b {|f - h|d\alpha < \varepsilon }$$ (ii)a continuous function $$g$$ on [a,b] with $${||g||_\infty}\leq{||f||_\infty}$$ such that $$\int\limits_a^b {|f - g|d\alpha < \varepsilon }$$ (Def:$$\space{R_\alpha }$$[a,b]=space of all Riemann-Stieltjes integrable functions w.r.t $$\alpha$$ on the interval [a,b])

• For (i), try functions of the form $h(x) = \frac{1}{N} \lfloor N f(x) \rfloor$, where $N$ is very large. Then $f$ and $h$ differ by less than $1/N$. Then modify this to be continuous for part (ii) while minimizing the change in the value of the integral. – Connor Harris Nov 16 '18 at 17:02