Workshop 結束

其實原名為 workshop in real analysis，最後 workshop 講唔到 integration，時間不足。我地有 7 次 meeting，每次用一至一個半鐘黎做 present，10 至 15 分鐘休息及將淨低嘅時間用黎講 notes。每次 workshop 為時三個鐘。已講 topic:

notes 喺每次 workshop 前準備，打下打下都 40 頁 notes。都整得幾辛苦，d theorem 用就用得多，從新 develop 返出黎都幾麻煩。any way 對我黎講得著唔係太多 :)，因為 d 野都比較 basic。希望下個 summer 會有人幫我完成埋 integration 果 part 同埋 expand 返第一課嘅野 (compactness, total boundedness, completeness, connectedness, etc)。

無聊之下嘅產物

喺 Royden (4th edition) p.452 有一 Theorem 將 locally compact Hausdorff space 有嘅 property 集埋一齊：

其中證明 (iii) 嘅主要工具係 Urysohn's lemma，但由 Urysohn's lemma 又可推出 Tietze Extension Theorem， 從 (iii) 我地應該可以做得更多。

Modification of (iii). Let $  X$ be locally compact Hausdorff. If $  \mathcal O$ is a neighborhood of a compact subset $  K$ of $  X$, then the continuous function $  f:K\to \mathbb R$ may be extended to a function $  F\in C_c(X)$ for which $  F$ vanishes outside $  \mathcal O$.
If $  f$ is bounded, say $  |f|\leq M$ for some $  M>0$, then the extension above can be chosen so that $  |F|\leq M$ on $  X$.

Proof. As $  K$ is compact and $  \mathcal O\supseteq K$, by (ii) of theorem 7 above there is an open $  V$ such that $  K\subseteq V\subseteq \overline{V}\subseteq \mathcal O$ with $  \overline{V}$ compact. Once again by (ii) of  theorem 7 above there is an open $  U$ such that
$  K\subseteq U\subseteq \overline{U}\subseteq V\subseteq \overline{V}\subseteq \mathcal O$.

Now we do our extension, as $  \overline{V}$ is a compact Hausdorff space, $  f$ is continuous (w.r.t. subspace topology) on $  K$, by the Tietze extension theorem there is a $  \mathcal F\in C (\overline{V})$ such that $  \mathcal F|_{K}= f|_K$, we extend $  \mathcal{F}$ on $  X\setminus \overline{V}$ by defining $  \mathcal F|_{X\setminus \overline{V}}\equiv 0$. The change of $  \mathcal F$ between $  \overline{V}$ and $  X\setminus \overline{V}$ may not be continuous, we will try to ``smooth" this transition. As $  K$ is compact and $  U\supseteq K$, by (iii) of theorem 7 there is a $  \psi \in C_c(X)$ such that $  0\leq \psi \leq 1$, $  \psi=1$ on $  K$ and $  \psi=0$ on $  X\setminus U$. Now we claim that the product of continuous functions $  F:= \mathcal F \cdot \psi$ will do.

Clearly $  \mathop{\mathrm{supp}} F \subseteq \mathop{\mathrm{supp}} \psi$, hence $  F$ has compact support. To show $  F$ is continuous on $  X$ we use the following fact:

Fact. Let $  X=\cup_\alpha X_\alpha$ be a union of open subsets. Then $  f:X\to Y$ is continuous if and only if the restrictions $  f|_{X_\alpha}:X_\alpha \to Y$ are continuous, where $  X_i$ has the subspace topology.

Proof. It follows from the observation that: For any subset $  A$ of $  Y$, $  f^{-1}(A)= \cup_{\alpha} (f|_{X_\alpha})^{-1}(A)$.$\qed$

Observe that both $  X_1:=V$ and $  X_2:=X\setminus \overline{U}$ are open, $  X=X_1\cup X_2$. It is enough to argue $  F|_{X_i}$'s are continuous. On $  X_1$, since $  \mathcal F$ is a continuous function on $  \overline{V}$, $  \mathcal F|_V$ is therefore a continuous function on $  V$. And as $  \psi$ is continuous on $  X$, so $  F|_{X_1}$ is continuous. On $  X_2$, since $  \psi|_{X\setminus U}\equiv 0$ $  \implies$ $  \psi|_{X\setminus \overline{U}} \equiv 0$, and thus $  F|_{X\setminus \overline{U}} \equiv 0$, hence $  F|_{X_2}$ is continuous on $  X_2$. We also note that $  F|_{X\setminus \mathcal O}\equiv 0$.

When $  |f|\leq M$ on $  K$, we repeat the proof above but that time $  \mathcal F$ can be chosen such that $  |\mathcal F|\leq M$ by the following version of Tietze extension theorem.$\qed$

建立呢種 extension 嘅原因係為左證明 Lusin's theorem on $  (X,\mathcal B(X),\mu)$，其中 $  X$ 為 locally compact Hausdorff，$  B(X)$ 為 Borel $  \sigma$-algebra on $  X$ 及 $  \mu$ 為 Radon measure (Royden's definition: A Borel measure such that Borel set is outer regular and open set is inner regular)，我嘅 approach (某習題) 係先證明 Lusin's theorem 對 simple function 成立，從而利用 simple functions $  \{\phi_n\}$, $  \phi_n\to f$ pointwise 及 Egoroff's theorem 及再利用上述 extension 完成證明 (已證明若 $  E\in \mathcal B(X),\mu(E)<\infty$，那麼 $  E$ 是 inner regular)。

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Some problem for entertainment:
Problem. Let $  f:\mathbb{R}\to\mathbb{R}$ be a differentiable function so that $  \displaystyle\left|f(x)-\sin(x^2)\right|\le\frac{1}{4}$ for any $  x\in\mathbb{R}$. Prove that there exists a sequence of real numbers $  \{x_n\}_{n=1}^\infty$ for which $  \lim_{n\to\infty} f'(x_n)=+\infty$ .

見工失敗

前排去左樂善堂余近卿中學面試去做份做八日就賺到五千嘅暑期班，貌似係為考試唔合格嘅學生而設嘅補底班黎 (亦即係我冇乜機會發揮嘅班)。咁好喇我自己又冇乜點見過工，見步行步。去到自我介紹，我講唔夠一分鐘就收口，深知不妙，最終不獲聘收埸。

我估最大原因係我冇乜教學經驗。原因係，首先我係該校校友推薦；其次係請人果位老師係 UST 師姐；再者，我個樣都算平易近人丫...。但間間學校都係以經驗為優先嘅話咁我邊鬼有經驗喎...。同埋我都唔算冇教學經驗，不過對像係班 UG year 1 ...。