NoteNextra-origin/content/Math4121/Math4121_L14.md

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

\omega(f,I) = \sup_I f - \inf_I f

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

\sum_{i\in \mathcal{P}}\Delta x_i<\epsilon

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

\omega(f,c) = \inf_{c\in I}\omega(f,I)

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

c_e(S) = \inf_{c\in C_s}\ell(C)

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.