TMA4225: Exercises for week 36

Look at the definition of the sum of a nonnegative function over a set on p.6 of the notes, and then prove the statements given there as Fact 2.2.

Show that, if \(\langle x_n\rangle_{x\in\mathbb{N}}\) is a sequence of nonnegative numbers and \(s\colon \mathbb{N}\to\mathbb{N}\) is a bijective mapping (i.e., a permutation), then \[\sum_{n=1}^\infty x_{s(n)}=\sum_{n=1}^\infty x_{n}.\] Use this to show that rearranging (by which we mean permuting the indices as above) an absolutely convergent sequence does not change its sum. (Recall that a conditionally convergent sequence can be rearranged to yield any sum whatsoever, or to diverge.)

Show that any sequence of extended real numbers (i.e., members of \([-\infty,\infty]\)) has a subsequence with a limit in \([-\infty,\infty]\).

Consider the function defined on \([0,1]\) by \[ f(x)=\begin{cases}0&x\notin\mathbb{Q},\\\frac1q&x=\frac pq\end{cases} \] where \(p\) and \(q\) in the second case are assumed to be nonnegative integers with no common factor.

Prove directly, using Riemann's criterion, that \(f\) is Riemann integrable. (Hint: Note that \(0\le f(x)\le1\) for all \(x\), and that \(\{x\colon f(x)\ge d\}\) is a finite set for each \(d\gt0\).)

The variation of a function f over a set A is \[ \mathop{\mathrm{var}}(f;A)=\sup\{f(x)\colon x\in A\}-\inf\{f(x)\colon x\in A\}. \] and the strength of discontinuity of f at a point x is \[ \sigma(f;x)=\inf\{\epsilon>0\colon \exists\delta>0\;\forall y\colon |x-y|<\delta\Rightarrow|f(x)-f(y)|<\epsilon\} \] Show that \[ \sigma(f;x)\le\inf_{\delta\gt0}\mathop{\mathrm{var}}(f;(x-\delta,x+\delta))\le 2\sigma(f;x). \]

Consider a bounded function \(f\colon[a,b]\to\mathbb{R}\), and let \(D\subseteq[a,b]\) be set of points where \(f\) is discontinuous. Assume that \(f\) is Riemann integrable, and show that \(\theta D=0\).

(Hint: When \(d\gt0\), define \(D_d=\{x\in[a,b]\colon \sigma(f;x)\gt d\}\) and use Riemann's criterion and problem 4 to show that \(\theta D_d=0\). Also note that \(D=\bigcup_{n\in\mathbb{N}} D_{1/n}\).)