84 lines
3.0 KiB
Markdown
84 lines
3.0 KiB
Markdown
# Math 4121 Lecture 15
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## Continue on patches for Riemann integral
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### Hankel's conjecture
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If $f$ is pointwise discontinuous (set of points of continuity is dense), then $f\in\mathcal{R}$.
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Why the conjecture is believable:
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- $\{w(f)\leq \sigma\}=\bigcup_{a\in D} I_a$, so then $S_{\sigma}$ contains no intervals, so the outer content _maybe_ zero.
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However, it turns out that the complement of the set of points of continuity can be large.
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#### Definition: Accumulation point
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Given a set $S$, $x$ is an accumulation point of $S$ if every open interval containing $x$ also contains infinitely many points of $S$.
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The derived set of $S$, denoted $S'$, is the set of all accumulation points of $S$.
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#### Lemma
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$$
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c_e(S)\leq c_e(S')
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$$
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Proof:
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Given $\epsilon > 0$, we can find open intervals $I_1, I_2, \ldots$ such that $S'\subseteq \bigcup_{i=1}^{n} I_i$ and
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$$
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\sum_{i=1}^{n} \ell(I_i) \leq c_e(S') + \frac{\epsilon}{2}
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$$
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So $S\setminus \bigcup_{i=1}^{n} I_i$ contains only finitely many points, say $N$, so we can cover $S\setminus \bigcup_{i=1}^{n} I_i$ by $N$ intervals of length $\frac{\epsilon}{2N}$. We call these intervals $J_1, J_2, \ldots, J_N$. Then $S$ is covered by the intervals $C=\bigcup_{i=1}^{n} I_i \cup \bigcup_{i=1}^{N} J_i$. and $\ell(C)\leq c_e(S') + \frac{\epsilon}{2} + \frac{N\epsilon}{2N}=c_e(S')+\epsilon$.
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So $c_e(S)\leq \ell(C)\leq c_e(S')+\epsilon$.
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EOP
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#### Corollary: sef of first species
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There exists $n\in \mathbb{N}$ such that $S^{(n)'}=\phi$.
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If $S$ is of the first species, then $c_e(S)=0$. This leads to further credence to Hankel's conjecture, but it begs the question is the complement of $\bigcup_{a\in D} I_a$ indeed of first species?
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#### Theorem (Baire Category Theorem)
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Every open set in $\mathbb{R}$ is a countable union of disjoint open intervals.
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Proof:
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Let $U$ be open and for each $t\in U$, set $a=\inf \{x:(x,t]\subseteq U\}$ and $b=\sup \{x:[t,x)\subseteq U\}$.
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We define $I(t)=(a,b)$, $U\subseteq \bigcup_{t\in U} I(t)$.
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Notice that if $I(t)\cap I(s)\neq \emptyset$, then there exists rational $r\in U$ such that $I(t)\cap I(s)\subseteq (r,r)$.
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> Take a dense, contable set in $[0,1]$, say $D=\{a_n\}_{n=1}^{\infty}$. Let $\epsilon>0$ and define $I_n=(a_n-\frac{\epsilon}{2^{n+1}}, a_n+\frac{\epsilon}{2^{n+1}})$. Can the entire interval $[0,1]$ be placed inside a union of intervals whose lengths add up to $\epsilon$?
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$$
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\sum_{n=1}^{\infty} \ell(I_n) = \sum_{n=1}^{\infty} \frac{2\epsilon}{2^{n+1}} = \epsilon
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$$
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_Harnack believed that the complement of a countable union of intervals is another countable union of intervals._
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Borel found the flaw in Harnack's argument. In fact, $[0,1]\setminus \bigcup_{n=1}^{\infty} I_n$ is uncountable.
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#### Theorem (Heine-Borel)
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If $\{U_\alpha\}_{\alpha\in A}$ is a collection of open sets (countable or uncountable) such that
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$$
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[a,b]\subseteq \bigcup_{\alpha\in A} U_\alpha
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$$
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then there exists a finite or countable subcollection $\{U_{\alpha_n}\}_{n=1}^{\infty}$ such that
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$$
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[a,b]\subseteq \bigcup_{n=1}^{\infty} U_{\alpha_n}
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$$
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Continue on next lecture.
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