First Problem set for 18.155, Fall 2002
Due September 17 in class or 2-174.

Not that the books of Folland [1] and Rudin [2] cover most of the material in the early lectures.

Some of these questions are very easy.

Problem 1   Prove the Partition of unity lemma written up in class on Tuesday September 20:

Lemma 0.1   [See also the lecture notes] Suppose $U_i,$ $i=1, \ldots , N$ is ,a finite collection of open sets in a locally compact metric space and $K \Subset \bigcup^N_{i=1} U_i$ is a compact subset, then there exist continuous functions $f_i \in C(X)$ with $0 \leq f_i \leq 1$, $\operatorname{supp} (f_i) \Subset U_i$ and

$$\sum_i f_i = 1 \hbox{ in a neighborhood of } K \, .$$ (1)

Hint(s). All functions here are supposed to be continuous, I just don't bother to keep on saying it.

1. Recall, or check, that the local compactness of a metric space $X$ means that for each point $x\in X$ there is an $\epsilon >0$ such that the ball $\{y\in X;d(x,y)\le\delta\}$ is compact for $\delta\le\epsilon.$
2. First do the case $n=1,$ so $K\Subset U$ is a compact set in an open subset.
   1. Given $\delta >0,$ use the local compactness of $X,$ to cover $K$ with a finite number of compact closed balls of radius at most $\delta.$
   2. Deduce that if $\epsilon >0$ is small enough then the set $\{x\in X;d(x,K)\le\epsilon\},$ where
      $$d(x,K)=\inf\limits_{y\in K}d(x,y),$$
      is compact.
   3. Show that $d(x,K),$ for $K$ compact, is continuous.
   4. Given $\epsilon >0$ show that there is a continuous function $g_\epsilon:\mathbb{R}\longrightarrow [0,1]$ such that $g_\epsilon(t)=1$ for $t\le\epsilon/2$ and $g_\epsilon(t)=0$ for $t>3\epsilon/4.$
   5. Show that $f=g_\epsilon \circ d(\cdot,K)$ satisfies the conditions for $n=1$ if $\epsilon >0$ is small enough.
3. Prove the general case by induction over $n.$
   1. In the general case, set $K'=K\cap U_1^\complement$ and show that the inductive hypothesis applies to $K'$ and the $U_j$ for $j>1;$ let $f'_j,$ $j=2,\dots,n$ be the functions supplied by the inductive assumption and put $f'=\sum_{j\ge2}f'_j.$
   2. Show that $K_1=K\cap\{f'\le\frac12\}$ is a compact subset of $U_1.$
   3. Using the case $n=1$ construct a function $F$ for $K_1$ and $U_1.$
   4. Use the case $n=1$ again to find $G$ such that $G=1$ on $K$ and $\operatorname{supp}(G)\Subset\{f'+F>\frac12\}.$
   5. Make sense of the functions
      $$f_1=F\frac{G}{f'+F},\ f_j=f_j'\frac{G}{f'+F},\ j\ge2$$
      and show that they satisfies the inductive assumptions.

Problem 2   Show that $\sigma$-algebras are closed under countable intersections.

Problem 3   Show that if $\mu$ is a complete measure and $E\subset F$ where $F$ is measurable and has measure 0 then $\mu(E)=0.$

Problem 4   The Borel $\sigma$-algebra is the smallest $\sigma$-algebra on a topological space containing the open sets; the elements of the Borel $\sigma$-algebra are called Borel sets.

1. Explain why such a smallest $\sigma$-algebra exists.
2. Show that compact sets are Borel sets.