8.3 A Non-integrable Function

We will now give an example of a function that is not integrable. Let

$$S=\{ {m\over n}\colon m\in\mbox{{\bf Z}},n\in\mbox{${\mbox{{\bf Z}}}^{+}$},\;\; m \mbox{ and } n \mbox{ are both odd}\}$$

$$T=\{ {m\over n}\colon m\in\mbox{{\bf Z}},n\in\mbox{${\mbox{{\bf Z}}}^{+}$},\;\; m \mbox{ is even and } n \mbox{ is odd}\}.$$

Then $S\cap T=\emptyset$, since if $${ {m\over n}={p\over q}}$$ where $m,\; n,$ and $q$ are odd and $p$ is even, then $mq=np$ which is impossible since $mq$ is odd and $np$ is even.

8.35   Lemma. Every interval $(c,d)$ in $\mbox{{\bf R}}$ with $d-c>0$ contains a point in $S$ and a point in $T$.

Proof: Since $d-c>0$ we can choose an odd integer $n$ such that $${ n>{3\over{d-c}}}$$, i.e., $nd-nc>3$. Since the interval $(nc,nd)$ has length $>3$, it contains at least two integers $p,q$, say $nc<p<q<nd$. If $p$ and $q$ are both odd, then there is an even integer between them, and if $p$ and $q$ are both even, there is an odd integer between them, so in all cases we can find a set of integers $\{r,s\}$ one of which is even and the other is odd such that $nc<r<s<nd$, i.e., $${c<{r\over n}<{s\over n}<d}$$. Then $${{r\over n}}$$ and $${{s\over n}}$$ are two elements of $(c,d)$ one of which is in $S$, and the other of which is in $T$. $\diamondsuit$

8.36   Example (A non-integrable function.) Let $D\colon [0,1]\to\mbox{{\bf R}}_{\geq 0}$ be defined by

$$D(x)=\cases{ 1 &if $x\in S$\cr 0 &if $x\notin S$.\cr}$$ (8.37)

I will find two partition-sample sequences $(\{P_n\},\{S_n\})$ and $(\{P_n\},\{T_n\})$ such that

$$\{\sum (D,P_n,T_n)\}\to 0$$

and

$$\{\sum (D,P_n,S_n)\}\to 1.$$

It then follows that $D$ is not integrable. Let $P_n$ be the regular partition of $[0,1]$ into $n$ equal subintervals.

$$P_n=\Big\{ 0,{1\over n},{2\over n},\cdots ,1\Big\}.$$

Let $S_n$ be a sample for $P_n$ such that each point in $S_n$ is in $S$ and let $T_n$ be a sample for $P_n$ such that each point in $T_n$ is in $T$. (We can find such samples by lemma 8.35.) Then for all $n\in\mbox{${\mbox{{\bf Z}}}^{+}$}$

$$\sum(D,P_n,S_n)=\sum_{i=1}^n D(s_n)(x_i-x_{i-1})=1$$

and

$$\sum(D,P_n,T_n)=\sum_{i=1}^n D(t_n)(x_i-x_{i-1})=0.$$

So $\lim\{\sum(D,P_n,S_n)\}=1$ and $\lim\{\sum(D,P_n,T_n)\}=0$. Both $(\{P_n\},\{S_n\})$ and $(\{P_n\},\{T_n\})$ are partition-sample sequences for $[0,1]$, so it follows that $D$ is not integrable.

Our example of a non-integrable function is a slightly modified version of an example given by P. G. Lejeune Dirichlet (1805-1859) in 1837. Dirichlet's example was not presented as an example of a non-integrable function (since the definition of integrability in our sense had not yet been given), but rather as an example of how badly behaved a function can be. Before Dirichlet, functions that were this pathological had not been thought of as being functions. It was examples like this that motivated Riemann to define precisely what class of functions are well enough behaved so that we can prove things about them.