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In this answer the following theorem is proved:

Theorem: Let $\alpha:[a, b] \to\mathbb {R} $ be a function such that the Riemann-Stieltjes integral $$\int_{a} ^{b} f(x) \, d\alpha(x) $$ exists for all continuous functions $f:[a, b] \to\mathbb {R} $. Then $\alpha $ is of bounded variation on $[a, b] $.

However the linked answer uses Banach Steinhaus theorem to prove it.

Is there any proof which does not use theorems from functional analysis and is rather based on the usual theorems dealing with Riemann-Stieltjes integration?

Paramanand Singh
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    I doubt such a proof exists. The setting is the natural habitat of functional analysis: a linear functional (integrating against some measure) defined on a normed space (the space of all continuous functions on $[a,b]$). – uniquesolution Apr 02 '19 at 11:18
  • I wonder if for “simple” concrete examples like $\alpha(x) = x \sin(1/x)$ it is possible to find a “concrete” function $f$ such that $\int_0^1 f(x) d\alpha(x)$ does not exist. – Martin R Jan 15 '23 at 16:33
  • @MartinR: I had forgotten about this question. It appears to be handled elsewhere on mathse: https://math.stackexchange.com/a/2427978/72031 – Paramanand Singh Jan 16 '23 at 00:55
  • @ParamanandSingh: That is a very nice answer, thanks for the response and the link! – Martin R Jan 16 '23 at 05:39
  • Here is another one https://math.stackexchange.com/q/1864023/42969 (and the answer looks wrong to me). What do you suggest: Close that one as a duplicate, or as off-topic (missing context)? – Martin R Jan 16 '23 at 12:23
  • @MartinR: the best would be to close for lack of context and add a comment linking the right dupe target. In case you vote to close as off topic I will also cast the vote to close it. – Paramanand Singh Jan 16 '23 at 14:39

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