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When defining the Riemann integral, say of $\int_a^b f(x)dx$ for $f:\mathbb{R}\to \mathbb{R}$ (or a more general setting), we take it as a limit of the sum of shrinking rectangles for example https://en.wikipedia.org/wiki/Riemann_integral . These rectangles are vertical. Is there also a definition taking horizontal rectangles?

For instance would this be an equivalent definition : Let $P_n$ be the a partition of $[a,b]$, i.e $x_1,x_2,x_3,\ldots x_n$, and $|P_n|=\max_{i=1,...,n}|x_{i+1}-x_i|$. Then define as the Riemann integral : $$\int_a^b f(x)dx= \lim_{|P_n|\to 0} \sum_{i=1}^n (f(x_{i+1})-f(x_i)) x_i. $$

Can we also do this for Riemann-Stieltjies integrals? Does anyone have a source?

joedondonjoe
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  • The Riemann-Stieltjes integral $\int_a^b f,dg$ can be obtained as a limit of sums with partition norm tending to $0$ only under special conditions. For example if $g$ is increasing and either $f$ or $g$ is continuous. See here. – RRL Jan 05 '20 at 23:59
  • When people explain the Lebesgue integral, they usually draw a certain picture that has horizontal rather than vertical strips. (It's a different picture than what you suggested.) While that is a very neat idea, I think the true secret sauce behind the Lebesgue integral is the way that measure is defined. – littleO Jan 06 '20 at 00:26

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For a partition $P=\{a=x_0, x_1,\ldots, x_{n-1}, x_n=b\}$ of $[a,b]$ and a non-decreasing function $g$ on $[a,b]$, define the upper Darboux sum of $f$ with respect to $g$ by $$ U(P,f,g) = \sum_{i=1}^n(g(x_i)-g(x_{i-1}))\sup_{x\in[x_{i-1},x_i]}f(x) $$ and the lower Darboux sum of $f$ with respect to $g$ by $$ L(P,f,g) = \sum_{i=1}^n(g(x_i)-g(x_{i-1}))\inf_{x\in[x_{i-1},x_i]}f(x). $$ Then the Riemann-Stieltjes integral of $f$ with respect to $g$ exists iff for all $\varepsilon>0$ there exists a partition $P$ such that $$ U(P,f,g) - L(P,f,g) < \varepsilon. $$ If in addition we have that $f$ or $g$ is continuous, this is equivalent to $$ \lim_{\|P\|\to 0}(U(P,f,g)-L(P,f,g)) = 0. $$

Math1000
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