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Let's say $f$ and $g$ are „nice enough” to $$\sum_{k}f\left(\xi_{k}\right)\left(g\left(x_{k+1}\right)-g\left(x_{k}\right)\right)\rightarrow\int_{a}^{b}f\left(x\right)dg\left(x\right),$$ where $\xi_{k}\in\left[x_{k},x_{k+1}\right]$, $a\leq x_{0}\leq x_{1}\leq\ldots\leq x_{n}\leq b$, but how do we interpret the convergence above? In what sense does it converge?

Kapes Mate
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    The usual limit definition for a sequence of real numbers. – calc ll Aug 09 '22 at 08:57
  • Anyway, is there any other definition or any other "type" for convergence of real numbers? Because I know there are a lot of "type of convergence" when we consider functions and sequences of functions... I think I mixed these two different things, so my original question doesn't really make sense... – Kapes Mate Aug 09 '22 at 09:34
  • There's more general notions like filters and nets, not sure what use they are here though. Do you mean like convergence in different topologies? – calc ll Aug 09 '22 at 10:51
  • In the case $g = 1$, the convergence is pointwise, and moreover it holds for arbitrary partition $a \leq x_0 \leq \dots \leq b$ with mesh size going to $0$ and arbitrary choices of tags $\xi_k \in [x_k, x_{k + 1}]$. This is Darboux's theorem. See theorem 4.2.4 on page 132 of https://mtaylor.web.unc.edu/wp-content/uploads/sites/16915/2018/04/anal1v.pdf. – Mason Aug 09 '22 at 22:16
  • See related https://math.stackexchange.com/a/2047959/72031 – Paramanand Singh Aug 10 '22 at 01:50

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