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?

  • $\begingroup$ I think there's an exercise on that on Baby Rudin. $\endgroup$ – dfnu Oct 14 '19 at 8:44
  • 1
    $\begingroup$ Thank you for the reminder! I found Question 6.12 and used the hint. $\endgroup$ – mathnoob777 Oct 15 '19 at 0:18

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