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I've been working on some problems involving Riemann integrals, and I been having problems relating $sup\{\mathcal{L}(\mathcal{P},f)\}$ to some other known quantity, like $\mathcal{L}(\mathcal{P},f)$ for some particular partition $\mathcal{P}$.

One of the specific problems I've been working on is showing that $\lim_{n \to \infty} \frac{1}{n}\sum_{i=1}^{n}f(\frac{i}{n}) = \int\limits_0^1f(x)d(x)$

Using only the definition of Riemann integral, I have gotten to the point where I have set up an inequality:

$\lim_{n \to \infty} \frac{1}{n}\sum_{i=1}^{n}m_i \leq \lim_{n \to \infty} \frac{1}{n}\sum_{i=1}^{n}f(\frac{i}{n}) \leq \lim_{n \to \infty} \frac{1}{n}\sum_{i=1}^{n}M_i$,

Where $m_i$ & $M_i$ are the local infimum and supremum of the function in the $i$th interval. The problem I'm running into is getting the $sup\{\mathcal{L}(\mathcal{P},f)\}$ and $inf\{\mathcal{U}(\mathcal{P},f)\}$ related to these quantities, because the sup and inf of the lower and upper sums are for all partitions, not just the particular one.

The only idea I've had so far (after many missteps), is to try and use the Archimedean principle to find some $n_0$ such that $sup\{\mathcal{L}(\mathcal{P},f)\} + \epsilon \geq \frac{1}{n_0}\sum_{i=1}^{n_0}m_i$, and then let $\epsilon \to 0$, but if anyone has any insight they can offer on how to be thinking about relating $sup\{\mathcal{L}(\mathcal{P},f)\}$ to some particular $\mathcal{P}$ used in proving something like this, I would be much obliged.

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  • $\begingroup$ I think it would help if you could specifically mention at least one or two problems you are thinking about related to this. Otherwise, it is not clear to me how exactly one would attempt to answer this question. (Also, it is easier to appreciate a technique when applied to solve a specific problem.) $\endgroup$ – Amitesh Datta Apr 17 '14 at 1:35
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    $\begingroup$ Is the example I provided not clear? I've been working on it for so long that it's all I can see, but if it is unclear I would be happy to add another one. $\endgroup$ – epsilonics Apr 17 '14 at 1:46
  • $\begingroup$ Dear @user47937, I apologise - I didn't notice the example you posted in your question. I attempted to answer your question below - please let me know if you're still stuck. (The hint is to use (i) in my answer combined with your reasoning in order to solve Exercise 2 - your question. You do need continuity of $f$ to solve the specific problem you're working on - Exercise 3 below provides a counterexample, otherwise.) $\endgroup$ – Amitesh Datta Apr 17 '14 at 1:58
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A couple of specific calculations (notation: denote by $U(f)$, the infimum over all partitions $P$ of $U(P,f)$; and denote by $L(f)$, the supremum over all partitions $P$ of $L(P,f)$. Here, $f:[a,b]\to \mathbb{R}$ is a continuous function):

Exercise 1: Prove that if $P,Q$ are partitions of $[a,b]$, then $L(P,f)\leq U(Q,f)$. (Hint: consider the common refinement $P\cup Q$ of $P$ and $Q$.) Conclude that $L(P,f)\leq L(f)\leq U(f)\leq U(P,f)$ for all partitions $P$ of $[a,b]$.

(i) A continuous function $f:[a,b]\to \mathbb{R}$ is Riemann integrable

Proof. A continuous function $f:[a,b]\to \mathbb{R}$ is necessarily uniformly continuous (compactness of the interval $[a,b]$). Choose $\epsilon > 0$. By uniform continuity, there exists $n\in \mathbb{N}$ such that $\left|f(x)-f(y)\right| < \epsilon$ whenever $\left|x-y\right|<\frac{1}{n}$ for $x,y\in [a,b]$.

If $P$ is any partition of $[a,b]$, then $\left|U(P,f)-L(P,f)\right|< \frac{1}{n} \sum_{i=1}^{n} \epsilon = \epsilon$. Since $P$ is arbitrary, we conclude (by Exercise 1 above) that $\left|U(f)-L(f)\right|<\epsilon$ for all $\epsilon > 0$ - that is, $U(f)=L(f)$ and $f$ is Riemann integrable. Q.E.D.

Exercise 2: Now, using the technique of uniform continuity in the proof of (i), can you solve your problem?

Exercise 3: Define $f:[0,1]\to \mathbb{R}$ by the rule $f(x)=1$ if $x\in \mathbb{Q}$ and $f(x)=0$ if $x\not\in \mathbb{Q}$. Calculate $U(f)$, $L(f)$, and prove that $f$ is not Riemann integrable.

Definition A function $f:[a,b]\to \mathbb{R}$ is almost continuous if the set of points of discontinuity of $f$ has measure $0$.

Exercise 4 (challenge): Prove that a function $f:[a,b]\to\mathbb{R}$ is Riemann integrable if and only if it is bounded and almost continuous.

Hope this helps! Please let me know if you're still stuck and I'm happy to help further.

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  • $\begingroup$ Thank you very much! This was very helpful, and I think I have a better understanding of how to think about problems like this. $\endgroup$ – epsilonics Apr 17 '14 at 3:25
  • $\begingroup$ Great, @user47937! I'm very happy to have helped. Let me know if you would like to discuss these matters further at any time. $\endgroup$ – Amitesh Datta Apr 17 '14 at 4:25

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