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\title{ Math 5210, HW II \\ due March 04}
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\noindent
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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\noindent
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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\noindent
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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\noindent
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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\noindent
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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\noindent
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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\noindent
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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\noindent
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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