2
$\begingroup$

The problem: Suppose $a < b$. I want to show that the function $f: [a,b]\to\mathbb{R},\ f(x) = -2x$ is integrable on the interval $[a,b]$, using the fact that a bounded function is integrable, if and only if its lower and upper Riemann-integrals on the interval $[a,b]$ are the same: \begin{align*} \underline{\int_a^b} f &= \sup\left\{ s(P_1) \mid s(P_1) = \sum_{i=1 }^{n} m_i (x_i - x_{i-1})\right\}\\ &= \inf\left\{ S(P_2) \mid S(P_2) = \sum_{j=1 }^{k} M_j (x_j - x_{j-1})\right\}\\ &= \overline{\int_a^b} f, \end{align*}

where $m_i$ and $M_j$ are the infimum and supremum of given subintervals respectively. Obviously $f$ is bounded, so it makes sense to use the above theorem. Based on the definitions of lower- and upper Riemann sums $s(P)$ and $S(P)$, I get that I should choose the partitions $P_1 = \{x_i\}_{i=0}^{n}$ and $P_2 = \{x_j\}_{j=0}^{k}$ so that the sums are equal, but I'm having trouble coming up with appropriate ones. How should I go about this?

$\endgroup$
  • 1
    $\begingroup$ Is it obvious they should be equal? Can you not just show you can make the upper and lower as close as you like to each other? $\endgroup$ – snulty Feb 1 '18 at 14:30
  • $\begingroup$ Of course they cannot be made equal, please reread carefully your notes. $\endgroup$ – Did Feb 1 '18 at 14:34
  • $\begingroup$ @Did I'm using Trench's book: digitalcommons.trinity.edu/mono/7, where in Theorem 3.2.6 he uses the "="-sign between the upper and lower integrals. $\endgroup$ – SeSodesa Feb 1 '18 at 14:39
  • 1
    $\begingroup$ Yeah, except that the supremum over every lower-integral and the infimum over every upper-integral are equal, not the lower-integral and the upper-integral for a given partition. $\endgroup$ – Did Feb 1 '18 at 14:58
3
$\begingroup$

I assume that $m_i$ and $M_j$ are supremums of $f$ on a given subinterval? If so, you are talking about Darboux sums, not Riemann sums.

Hint:

  • You don't need to find partition for which the sums are equal, only similar (i.e. $S(P)-s(P)<\epsilon$).
  • You can easily calculate the values $M_j$ and $m_i$ since $f$ is a monotone function.
$\endgroup$
  • $\begingroup$ Alright, let me think about this for a while. $\endgroup$ – SeSodesa Feb 1 '18 at 14:35
  • $\begingroup$ Alright, so I could simply use a partition $P$ whose norm $\lVert P \rVert = \frac{b-a} n \to 0$ as $n\to\infty$, making it possible for the difference $S(P) - s(P)$ to approach zero, effectively making it less than $\varepsilon$. $\endgroup$ – SeSodesa Feb 1 '18 at 14:55
1
$\begingroup$

Compute the lower and the upper sum for an equidistant partition into $N\gg1$ parts exactly and verify that their difference can be made arbitrarily small by choosing $N$ large enough. Then use the squeeze theorem on the relation $$\underline{S}_N\leq\underline{\int_a^b} f(x)\>dx\leq\overline{\int_a^b} f(x)\>dx\leq\overline{S}_N\ .$$

$\endgroup$

Your Answer

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

Not the answer you're looking for? Browse other questions tagged or ask your own question.