Math4121 Lecture 14

Recap

Hankel developed Riemann’s integrability criterion

Definition: Oscillation

Given an interval $I\subset[a,b]$, $f:[a,b]\to\mathbb{R}$ the oscillation of $f$ at $I$ is

Theorem 2.5: Riemann’s Integrability Criterion

A bounded function $f$ is Riemann integrable if and only if for every $\sigma,\epsilon>0$ there exists a partition $P$ of $[a,b]$ such that

where $\mathcal{P}=\{i:\omega(f,I_i)>\sigma\}$.

Proof:

To prove Riemann’s Integrability Criterion, we need to show that a bounded function $f$ is Riemann integrable if and only if for every $\sigma, \epsilon > 0$, there exists a partition $P$ of $[a, b]$ such that the sum of the lengths of the intervals where the oscillation exceeds $\sigma$ is less than $\epsilon$.

QED

Proposition 2.4

For point $c\in[a,b]$, define the oscillation at $c$ as

Homework question 6: $f$ is continuous at $c$ if and only if $\omega(f,c)=0$.

So we can restate the previous theorem as:

Given $\sigma>0$, define $S_\sigma=\{c\in[a,b]:\omega(f,c)>\sigma\}$.

Restate the theorem as:

$f\in\mathscr{R}[a,b]$ if and only if for every $\sigma,\epsilon>0$ there exists intervals $I_1,I_2,\cdots,I_n$ such that $S_\sigma\subset\bigcup_{i=1}^{n}I_i$ and $\sum_{i=1}^{n}\ell(I_i)<\epsilon$. where $\ell(I)$ is the length of the interval $I$.

Definition: Outer content

Given a set $S$, a finite cover of $S$ is a collection of intervals $C=\{I_1,I_2,\cdots,I_n\}$ such that $S\subseteq\bigcup_{i=1}^{n}I_i$.

The length of the cover $C$ is $\ell(C)=\sum_{i=1}^{n}\ell(I_i)$.

The outer content of $S$ is

where $C_s$ is the set of all finite covers of $S$.

Example:

$S=\{x_1,\ldots,x_n\}$, then $c_e(S)=0$.

• Let $I_i=(x_i-\frac{\epsilon}{2n},x_i+\frac{\epsilon}{2n})$, so $\sum_{i=1}^{n}\ell(I_i)=\epsilon$

$S=\{\frac{1}{n}\}_{n=1}^{\infty}$, then $c_e(S)=0$.

• In this case, we can only use finite cover, however, there is only one “accumulation point”, so we can cover it with a single interval, and the remaining points can be covered by finite intervals. (for any $\epsilon>0$, we can construct a finite cover with length $\epsilon$ that covers all points.)

$S=\mathbb{Q}\cap[0,1]$, then $c_e(S)=1$.

• In this case, by covering the interval with $[0,1]$, we can get the length of the cover is at most 1.
• Suppose there exists a cover $C$ with $\sum_{I\in C}\ell(I)<1$, then there must be a gap in the intervals, however, since the $\mathbb{Q}$ is dense in $\mathbb{R}$, there must be a point in the gap, which is a contradiction.

Theorem 2.5: Hankel’s criterion for Riemann integrability

A function $f\in\mathscr{R}[a,b]$ if and only if $c_e(S_\sigma)=0$ for all $\sigma>0$.

The idea is that if the oscillation of a function can be bounded by a finite set that the total length is small, then the function is Riemann integrable.

Hankel’s idea was to apply this theorem to determining how discontinuous a function could be a Riemann integrable function.

A set $S$ is dense in $X$ if every point of $X$ is a limit point of $S$.

Definition: Totally discontinuous

$f$ is totally discontinuous if the points of continuity of $f$ are not dense.

For example, $f(x)=\begin{cases} 0 & x\in\mathbb{Q}\\ 1 & x\notin\mathbb{Q} \end{cases}$ is totally discontinuous.

Definition: Pointwise discontinuity

$f$ is pointwise discontinuous if they are dense in $[a,b]$.

Hankel’s conjecture: $f$ is pointwise discontinuous, then $f$ is integrable.