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I'm asked to show that

$\lim_{n\to\infty}\frac{b-a}{n}\sum_{k=1}^n f(a+k\frac{b-a}{n}) = \int_{a}^{b}f(x)dx$.

I tried to come up with a step function that is constant on each of the open intervals $(a+k\frac{b-a}{n},a+(k+1)\frac{b-a}{n})$ and has a jump size of $\frac{b-a}{n}$ at $a+k\frac{b-a}{n}$. I think $\alpha_n = [floor(\frac{x-a}{\frac{b-a}{n}})](\frac{b-a}{n})$ works, so I want to show that $\alpha_n$ converges pointwise to $x$ and then invoke the Helly's second theorem.

I don't know how to show that function converges to $x$, any help will be appreciated.

middlethird_cantor
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1 Answers1

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You're overthinking this problem. The definition of a Riemann-Stieltjes integral $\int_a^b{f(x)\,d\alpha(x)}$ is $$\int_a^b{f(x)\,d\alpha(x)}=\lim_{\substack{P\,\vdash[a,b]\\ P_{j-1}\leq x_j\leq P_j}}{f(x_j)(\alpha(P_j)-\alpha(P_{j-1}))}$$ where the limit is along the net of partitions of $[a,b]$ with arbitrary $x_j$ interleaved, partially ordered by subdivision of partitions. (If those last words sound like nonsense to you, I claim that they are a fancy way of saying roughly "take the mesh $\mu(P)\to0$" that avoids dealing with Darboux sums.)

So choose $P$ and $\{x_j\}_j$ wisely: $$P_n=\left\{a,a+\frac{b-a}{n},a+2\cdot\frac{b-a}{n},\dots,b\right\}$$ $$x(n)_k=a+k\cdot\frac{b-a}{n}\quad\forall k\in[0,n]$$

Jacob Manaker
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  • Convergence as the mesh of the partition $|P| \to 0$ may fail yet the function may be RS integrable in the sense that for every $\epsilon$ there exists a partition $P_\epsilon$ such that if $P$ is a refinement, i.e., $P_\epsilon \subset P$, then $|S(P,f,\alpha) - I| < \epsilon$ where $S(P,f,\alpha)$ is a tagged RS sum. For a counterexample see here. So there is not one definition. They will be equivalent if $\alpha$ is increasing and either $f$ or $\alpha$ is continuous. – RRL Apr 24 '19 at 04:22
  • Yes, the definition I give is the stronger (Riemann-Stieltjes-Young) definition used in that problem. I'll edit my answer to make it clear that my mesh aside is an oversimplification. – Jacob Manaker Apr 24 '19 at 04:32