Let $f:[0,1]\to \mathbb{R}$ be Darboux integrable. I ask for a proof of $$\int_0^1f=\lim_{n\to \infty}\frac1n\sum_{i=1}^{n}f\left(\frac in\right)$$ where the integral in the left hand side is the Darboux Integral of $f$. I know how to prove this when the integral is the Riemann Integral or when $f$ is monotone. I also know that the Darboux Integral is equivalent to the Riemann Integral. But can the above be proven directly without going into Riemann Sums?

For completeness here are the definitions: If $\mathcal{P}=\left\{a=x_0<...<x_n=b\right\}$ partitions $[a,b]$ and $f:[a,b]\to \mathbb{R}$ is bounded we define \begin{equation} M_i(f)=\sup_{x\in [x_{{i-1}},x_i]}f(x)\text{ and }m_i(f)=\inf_{x\in [x_{{i-1}},x_i]}f(x)\end{equation} The Darboux sums are: \begin{equation} U_{f,\mathcal{P}}:=\sum\limits_{i=1}^{n}{M_i(f)\left( x_i-x_{i-1} \right)}\text{ and } L_{f,\mathcal{P}}:=\sum\limits_{i=1}^{n}{m_i(f)\left( x_i-x_{i-1} \right)} \end{equation} The Darboux upper and lower integrals are \begin{equation} \overline{\int\limits_a^b}f:=\inf_{\mathcal{P}}U_{f,\mathcal{P}}\text{ and } \underline{\int\limits_{a}^{b}}f:=\sup_{\mathcal{Q}}L_{f,\mathcal{Q}}\end{equation} If the two coincide, $f$ is integrable and their common value is the Darboux integral of $f$.

We also have this criterion:

$f$ is integrable iff $\forall \epsilon>0$ there exists a partition $\mathcal{P}$ of $[a,b]$ such as that $U_{f,\mathcal{P}}-L_{f,\mathcal{P}}<\epsilon$

which can be written equivalently as

$f$ is integrable iff there exists a sequence of partitions $\mathcal{P}_n$ of $[a,b]$ such as that $U_{f,\mathcal{P}_n}-L_{f,\mathcal{P}_n}\to 0$. In that case \begin{equation} \int\limits_{a}^{b}f=\lim_{n\to \infty}U_{f,\mathcal{P}_n}=\lim_{n\to \infty}L_{f,\mathcal{P}_n}\end{equation}

Proof if $f$ is monotone: WLOG $f$ is increasing. Then \begin{equation}\mathcal{P}_n=\left\{ 0=x_0<...<x_i=\frac{i}{n}<...<x_n=1 \right\}\end{equation} partitions $[0,1]$. Obviously, $M_i(f)=f(\frac in)$ and so \begin{equation} \int\limits_{0}^{1}f=\lim_{n\to \infty}U_{f,\mathcal{P}_n}=\lim_{n\to \infty}\sum\limits_{i=1}^{n}{M_i(f)\left( x_i-x_{i-1} \right)}=\lim_{n\to \infty}\sum\limits_{i=1}^{n}{f(\frac in)\left(\frac1n\right)}\end{equation}

I will say it again: I don't want Riemann sums to be used, only upper and lower sums as above.

Thank you in advance.

  • 5
    $\begingroup$ What happens if you compare $f(i/n)$ to $M_i(f)$ and $m_i(f)$ for $x_i=i/n$? $\endgroup$ – Alex R. Jan 4 '13 at 19:41
  • 1
    $\begingroup$ @Alex I can't believe it's that simple. You have $m_i<f(\frac in)<M_i\implies L_{f,\mathcal{P}_n}<\sum_{i=1}^n\frac1n f(\frac in)<U_{f,\mathcal{P}_n}$ and by letting $n\to \infty$ we have the result. I believe this is correct. Why don't you post it as an answer? $\endgroup$ – Nameless Jan 4 '13 at 19:45
  • 1
    $\begingroup$ You are allowed to answer your own questions. You'll learn more than me if you do :). $\endgroup$ – Alex R. Jan 4 '13 at 20:04
  • 1
    $\begingroup$ @ThomasAndrews Yeah I mistook $\exists$ for $\forall$. Can you explain the point with the refinements? $\endgroup$ – Nameless Jan 4 '13 at 20:59
  • 1
    $\begingroup$ @ThomasAndrews Here is what I think. Let $\mathcal{P}_n$ be paritions so that $U_{f,P_n}-L_{f,P_n}\to 0$ and $\mathcal{Q}_n=P_n\cup\left\{0,\frac1n,...,1\right\}$. Then $U_{f,Q_n}-L_{f,Q_n}\to 0$. Can we show that $U_{f,Q_n}\ge \sum\limits_{i=1}^{n}{f(\frac in)\left(\frac1n\right)}$? $\endgroup$ – Nameless Jan 4 '13 at 21:14

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.