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\title{ Math 5210,  HW II \\ due March 04}
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1)  A metric space $X$ is separable if it contains a dense countable set $S$. Prove that any open set $V$ in $X$ is a union of 
balls centered at points in $S$ and with rational radii. (Since the set of such balls is countable, it follows that any open set is a countable 
union of balls). 
 
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2) Let $X=[0,1]^2$.  Choose the distance on $X$ wisely, and use the previous exercise to prove that any open set in $X$ is Lebesgue measurable. 




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3) Let $P=[0,1]^2$. If $E$ and $F$ are two elementary sets such that $E\cup F=P$ then $m(E\cap F)=m(E)+m(F)-1$.  
Now assume $E=\cup_{i=1}^{\infty} E_i$ and $F=\cup_{i=1}^{\infty} F_i$, disjoint unions of elementary sets each, and 
$E\cup F=P$. Observe that $E\cap F$ is the disjoint union of $E_i\cap F_j$.  Prove that 
\[ 
\sum_{i,j} m(E_i\cap F_j) = \sum_i m(E_i) + \sum_j m(F_j) -1. 
\] 

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4)  Let $\sum _{n=1}^{\infty} x_n$ be a series of non-negative real numbers. Show that its sum (which can be $\infty$) is equal to the supremum 
of the set of sums $\sum _{n\in S} x_n$ where $S$ runs over all finite subsets of the set of natural numbers. Conclude that any sequence 
of non-negative numbers can be added in any order. 

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5) In the following exercises, $\mathcal M$ is a $\sigma$-algebra of a non-empty set $X$, that is, a family of subsets of $X$ closed under complements and countable unions, and 
 $\mu$ is a $\sigma$-measure. Let $A_1 \supseteq A_2 \supseteq \ldots $ be a sequence of sets 
in $\mathcal M$. Let $A=\cap_{i=1}^{\infty} A_i$. Prove that $\lim_{i\mapsto \infty} \mu(A_i)= \mu(A)$, assuming that $\mu(X)=1$. 

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6) A subset of $X$ is called measurable if it belongs to $\mathcal M$.  Let $f: X \rightarrow \mathbb R$ prove that   
\[ 
\{ x | f(x) < c \} 
\] 
is measurable for every $c\in \mathbb R$ if and only if 
\[ 
\{ x | f(x) \leq  c \} 
\] 
is measurable  for every $c\in \mathbb R$. 

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7) Let $f_n : X\rightarrow \mathbb R$ be a sequence of measurable functions on $X$. Prove that 
\[ 
g(x)=\inf \{ f_1(x), f_2(x), \ldots \} \text{ and } G(x)=\sup \{ f_1(x), f_2(x), \ldots \}
\] 
are measurable functions. 


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8) Let $f$ be an integrable function on $X$, such that $f(x)\geq 0$ for all $x\in X$. 
Prove that $\int_X f=0$ if and only if the measure of $A=\{x\in X ~|~ f(x) >0\}$ is 0, that is, $f=0$ almost everywhere.
 Hint consider the sets $A_n=\{x\in X ~|~ f(x) >1/n\}$ for $n=1,2, \ldots$. 

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9) Let $X=(0,1]$, with the usual measure, and let $f(x)= 1/\sqrt{x}$. Use the monotone convergence theorem to prove that 
$f$ is integrable and compute its integral.    



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4) Let $f:[a,b] \rightarrow \mathbb R$ be a continuous function such that $f(x) \geq 0$ for all $x\in [a,b]$.  Prove that 
\[ 
\int_a^b f = 0 
\] 
implies $f(x)=0$ for all $x\in [a,b]$. 

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5) Let $(X,d)$ be a metric space. Let $(x_n)$ and $(y_n)$ be two Cauchy sequences in $X$. Prove that $(d(x_n,y_n))$ is a Cauchy sequence in $\mathbb R$. 

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6)  Let $K\subset \mathbb R$ be a set consisting of $0$ and all $1/n$, $n=1,2,3,\ldots$. Prove that $K$ is compact directly using the definition, i.e. every open cover has a 
finite subcover. 

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7)  Let $F_1 \supseteq F_2 \supseteq  \ldots $ be a descending sequence of non-empty compact subsets. Prove that $\cap_{n=1}^{\infty} F_n$ is non-empty. 

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8) Let $(X,d)$ be a metric space and $f_n$ a sequence of continuous functions  $f_n : X \rightarrow \mathbb R$ uniformly converging to $f$. Let $x_n$ be a 
sequence of points in $X$ such that $\lim_n x_n=x\in X$. Prove that $\lim_n f_n(x_n)=f(x)$ . 

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9) A subset $\mathbb R^n$ is convex if for any two points $x,y\in C$, the segment $[x,y]$ is contained in $C$. Prove that $C$ is connected. 


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