# Showing every Riemann-Stieljes integrable function can be approximated by a continuous function

Hi there I am preparing for a midterm and struggling to understand whether we can approximate any RJ-integrable function on a bounded interval using a continuous function:

Suppose $$α$$ is a (monotonically) increasing function in $$[a,b]$$ and $$f∈R([a,b],α)$$. Can we show that that for every $$ε>0$$, there exists some continuous function $$g$$ such that $$∫_a^b|f-g|dα<ε?$$

Before tonight I thought the answer was yes:

Using Definition 1 from Anevski's paper, since $$f$$ is RJ-integrable wrt $$\alpha$$, then for every $$ε>0$$, there are step-functions $$g_1$$ and $$g_2$$ such that $$g_1\le f \le g_2$$ and
$$∫_a^b (g_2-g_1) dα<ε.$$

And the RJ-integral of $$f$$ is $$∫_a^b f dα =\sup\{\int_a^b gd\alpha \text{ s.t. g\leq f, g step function}\}.$$

But I am no longer certain it could be this easy. If possible, would you please give me some advice on how to construct a solid proof?

• I think there's an exercise on that on Baby Rudin. – dfnu Oct 14 '19 at 8:44
• Thank you for the reminder! I found Question 6.12 and used the hint. – mathnoob777 Oct 15 '19 at 0:18