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])

  • $\begingroup$ 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. $\endgroup$ – Connor Harris Nov 16 '18 at 17:02

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