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Homework 10, due Wednesday, December 8th (Sections 3.5, 6.1) Solutions

• 1. Let $G\colon [a,b]\to [c,d]$ be a continuous increasing surjection.
  • (a) For a Borel set $E\subset [c,d]$, show that $m(E)=\mu_G(G^{-1}(E))$.
    [Hint: first consider when $E$ is an open then closed.]
  • (b) For $f\in L^1([c,d],\mathcal{B}_{[c,d]}, m)$, show that $$\int_{[c,d]} f\ dm = \int_{[a,b]} f\circ G\ d\mu_G.$$
  • (c) Suppose $G$ is absolutely continuous. Show that the above integrals also equal $\int_{[a,b]} (f\circ G) G'\ dm$.
• 2. We say $F\colon \mathbb{R}\to \mathbb{C}$ is Lipschitz continuous if there exists $M>0$ so that $|F(x) - F(y)|\leq M|x-y|$ for all $x,y\in \mathbb{R}$. Show that a function $F$ is Lipschitz continuous with constant $M$ if and only if $F$ is absolutely continuous and $|F'|\leq M$ $m$-almost everywhere.
• 3. For $(a,b)\subset \mathbb{R}$ (possibly equal), we say a function $F\colon (a,b)\to \mathbb{R}$ is convex if $$F(\lambda s + (1-\lambda) t) \leq \lambda F(s) + (1-\lambda) F(t)$$ for all $s,t\in (a,b)$ and $\lambda\in (0,1)$.
  • (a) Show that $F$ is convex if and only if for all $s,s', t,t'\in (a,b)$ satisfying $s\leq s' < t'$ and $s < t\leq t'$ one has $$\frac{F(t) - F(s)}{t-s}\leq \frac{ F(t') - F(s')}{t'-s'}.$$
  • (b) Show that $F$ is convex if and only if $F$ is absolutely continuous on every compact subinterval of $(a,b)$ and $F'$ is increasing on the set where it is defined.
  • (c) For convex $F$ and $t_0\in (a,b)$, show that there exists $\beta\in \mathbb{R}$ satisfying $F(t) - F(t_0) \geq \beta (t-t_0)$ for all $t\in (a,b)$.
  • (d) (Jensen's Inequality) Let $(X,\mathcal{M},\mu)$ be a measure space with $\mu(X)=1$. Suppose $g\in L^1(X,\mu)$ is valued in $(a,b)$ and $F$ is convex on this interval. Show that $$F\left( \int_X g\ d\mu \right) \leq \int F\circ g\ d\mu.$$ [Hint: use part (c) with $t_0=\int g\ d\mu$ and $t=g(x)$.]
• 4. Let $(X,\mathcal{M},\mu)$ be a measure space and let $0 < p < q < \infty$.
  • (a) Show that $L^p(X,\mu)\not\subset L^q(X,\mu)$ if and only if for all $\epsilon>0$ there exists $E\in \mathcal{M}$ with $0<\mu(E)<\epsilon$.
  • (b) Show that $L^q(X,\mu)\not\subset L^p(X,\mu)$ if and only if for all $R>0$ there exists $E\in \mathcal{M}$ with $R<\mu(E)<\infty$.
  [Hint: for the backwards direction of both parts, construct a function of the form $f= \sum a_n 1_{E_n}$ for a disjoint family $(E_n)_{n\in \mathbb{N}}\subset \mathcal{M}$.]
• 5. Let $(X,\mathcal{M},\mu)$ be a measure space. For a measurable function $f\colon X\to \mathbb{C}$, we say $z\in \mathbb{C}$ is in the essential range of $f$ if $$\mu(\{x\in X\colon |f(x) - z|<\epsilon\})>0$$ for all $\epsilon>0$. Denote the essential range of $f$ by $R_f$.
  • (a) Show that $R_f$ is closed.
  • (b) For $f\in L^\infty(X,\mu)$, show that $R_f$ is compact with $\|f\|_\infty = \max\{|z|\colon z\in R_f\}$.

Homework 9, due Wednesday, December 1st (Sections 3.4, 3.5) Solutions

• 1. Let $f\in L^1(\mathbb{R}^n,m)$ be such that $m(\{x\in \mathbb{R}^n\colon f(x)\neq 0\})>0$.
  • (a) Show that there exists $C,R>0$ so that $Hf(x)\geq C|x|^{-n}$ for $|x|>R$.
  • (b) Show that there exists $C'>0$ so that $m(\{x\in \mathbb{R}^n\colon Hf(x)> \epsilon\}) \geq \frac{C'}{\epsilon}$ for all sufficiently small $\epsilon>0$.
    [Note: this shows that the Hardy–Littlewood maximal inequality is sharp up to the choice of constant.]
• 2. For a Borel set $E\subset \mathbb{R}^n$, the density of $E$ at a point $x$ is defined as $$D_E(x):= \lim_{r\to 0} \frac{m(E\cap B(x,r))}{m(B(x,r))}$$ whenever the limit exists.
  • (a) Show that $D_E$ is defined $m$-almost everywhere and $D_E=1_E$ $m$-almost everywhere.
  • (b) For $0<\alpha<1$, find an example of an $E$ and so that $D_E(0)=\alpha$. [Hint: use a sequence of annuli.]
  • (c) Find an example of an $E$ so that $D_E(0)$ does not exist. [Hint: use another sequence of annuli.]
• 3. Let $\nu$ be a regular signed or complex Borel measure on $\mathbb{R}^n$ with Lebesgue decomposition $\nu=\lambda +\rho$ with respect to the Lebesgue measure $m$, where $\lambda\perp m$ and $\rho\ll m$. Show that $\lambda$ and $\rho$ are both regular. [Hint: first show $|\nu|=|\lambda|+|\rho|$.]
• 4. Define $$F(x):=\begin{cases}x^2 \sin(\frac1x) &\text{if }x\neq 0\\ 0 & \text{otherwise} \end{cases} \qquad\text{and}\qquad G(x):=\begin{cases}x^2 \sin(\frac{1}{x^2}) &\text{if }x\neq 0\\ 0 & \text{otherwise} \end{cases}.$$
  • (a) Compute $F'$ and $G'$.
  • (b) Show that $F\in BV([-1,1])$ but $G\not\in BV([-1,1])$.
• 5. Suppose $(F_n)_{n\in \mathbb{N}}\subset NBV$ converges pointwise to a function $F$. Show that $T_F\leq \displaystyle\liminf_{n\to\infty} T_{F_n}$.

Homework 8, due Wednesday, November 10th (Sections 3.2, 3.3) Solutions

• 1. For $j=1,2$, let $\mu_j$, $\nu_j$ be $\sigma$-finite measures on $(X_j,\mathcal{M}_j)$ with $\nu_j\ll \mu_j$. Show that $\nu_1\times \nu_2 \ll \mu_1\times \mu_2$ with $$\frac{ d(\nu_1\times \nu_2)}{d(\mu_1\times \mu_2)}(x_1,x_2) = \frac{d\nu_1}{d\mu_1}(x_1) \frac{d\nu_2}{d\mu_2}(x_2)$$ for $(\mu_1\times \mu_2)$-almost every $(x_1,x_2)\in X_1\times X_2$.
• 2. On $([0,1],\mathcal{B}_{[0,1]})$, let $m$ be the Lebesgue measure and let $\nu$ be the counting measure.
  • (a) Show that $m\ll \nu$, but $dm\neq f\ d\nu$ for any function $f$.
  • (b) Show that there does not exist $\lambda\perp m$ and $\rho\ll m$ so that $\nu=\lambda+\rho$.
  [Note: this shows the $\sigma$-finiteness assumption in the Lebesgue–Radon–Nikodym theorem is necessary.]
• 3. Let $(X,\mathcal{M},\mu)$ be a $\sigma$-finite measure space, and let $\nu$ be a $\sigma$-finite signed measure on $(X,\mathcal{M})$ with $\nu \ll \mu$.
  • (a) Show that $\left| \frac{d\nu}{d\mu} \right| = \frac{d|\nu|}{d \mu}$.
  • (b) Show that $\frac{d\nu}{d\mu}\in L^1(X,\mu)$ if and only if $\nu$ is finite.
  • (c) Suppose $\nu$ is positive and let $\lambda:=\nu+\mu$. Show that $0\leq \frac{d \nu}{d\lambda}<1$ $\mu$-almost everywhere and that $$\frac{d\nu}{d\mu} = \frac{\frac{d \nu}{d\lambda}}{1-\frac{d \nu}{d\lambda}}.$$
• 4. Let $\nu$ be a complex measure on $(X,\mathcal{M})$. Show that $\nu=|\nu|$ iff $\nu(X)=|\nu|(X)$.
• 5. Let $\nu$ be a complex measure on $(X,\mathcal{M})$. For $E\in \mathcal{M}$, show $$\begin{align*} |\nu|(E) &= \sup\left\{ \sum_{j=1}^n |\nu(E_j)| \colon E=E_1\cup\cdots \cup E_n \text{ is a partition}\right\}\\ &= \sup\left\{ \sum_{j=1}^\infty |\nu(E_j)| \colon E=\bigcup_{j=1}^\infty E_j \text{ is a partition}\right\}\\ &= \sup\left\{ \left|\int_E f\ d\nu \right| \colon |f|\leq 1\right\}. \end{align*}$$
• 6. Let $M(X,\mathcal{M})$ denote the set of complex measures on a measurable space $(X,\mathcal{M})$.
  • (a) Show that $\| \nu\|:=|\nu|(X)$ defines a norm on $M(X,\mathcal{M})$.
  • (b) Show that $M(X,\mathcal{M})$ is complete with respect to the metric $\|\nu-\mu\|$.
  • (c) Suppose $\mu$ is a $\sigma$-finite measure on $(X,\mathcal{M})$ and $(\nu_n)_{n\in \mathbb{N}} \subset M(X,\mathcal{M})$ satisfies $\nu_n\ll \mu$ for all $n\in \mathbb{N}$. For $\nu\in M(X,\mathcal{M})$, show that $\|\nu_n - \nu\|\to 0$ if and only if $\nu\ll \mu$ and $\frac{d\nu_n}{d\mu}\to \frac{d\nu}{d\mu}$ in $L^1(\mu)$.

Homework 7, due Wednesday, November 3rd (Sections 2.5, 3.1) Solutions

• 1. On $([0,1],\mathcal{B}_{[0,1]})$, let $m$ be the Lebesgue measure and let $\nu$ be the counting measure. For the diagonal set $$D:=\{(t,t)\in [0,1]^2\colon 0\leq t\leq 1\},$$ show that $\iint 1_D\ dmd\nu$, $\iint 1_D\ d\nu dm$, and $\int 1_D\ d(m\times \nu)$ are all distinct.
  [Note: this shows the $\sigma$-finiteness assumption in the Fubini–Tonelli theorem is necessary.]
• 2. Let $(X,\mathcal{M},\mu)$ be a $\sigma$-finite measure space and $f\in L^+(X,\mu)$.
  • (a) Show that $$G_f:=\{(x,t)\in X\times [0,\infty)\colon t\leq f(x)\}$$ is $\mathcal{M}\otimes \mathcal{B}_{\mathbb{R}}$-measurable with $\mu\times m(G_f) = \int_X f\ d\mu$.
  • (b) Prove the layer cake formula: $$\int_X f\ d\mu = \int_{[0,\infty)} \mu(\{x\in X\colon f(x)\geq t\})\ dm(t).$$
  • (c) Use part (a) and continuity from below to give an alternate (albeit circular) proof of the monotone convergence theorem.
• 3. Suppose $f\in L^1( (0,1),m)$, and define $$g(x):=\int_{(x,1)} \frac{1}{t} f(t)\ dm(t) \qquad 0 < x < 1.$$ Show that $g\in L^1((0,1),m)$ with $\int_{(0,1)} g\ dm = \int_{(0,1)} f\ dm$.
• 4. Let $\nu$ and $\mu$ be signed measures on a measurable space $(X,\mathcal{M})$.
  • (a) Show that $E$ is $\nu$-null if and only if $|\nu|(E)=0$.
  • (b) Show the following are equivalent:
    • (i) $\nu\perp \mu$
    • (ii) $|\nu|\perp \mu$
    • (iii) $\nu^+\perp \mu$ and $\nu^-\perp \mu$.
• 5. Let $\nu$ be a signed measure on a measurable space $(X,\mathcal{M})$.
  • (a) Show $L^1(X,\nu) = L^1(X,|\nu|)$.
  • (b) For $f\in L^1(X,\nu)$, show $|\int_X f\ d\nu|\leq \int_X |f|\ d|\nu|$.
  • (c) For $E\in \mathcal{M}$, prove the following formulas:
    • (i) $\nu^+(E)= \sup\{\nu(F)\colon F\subset E, F\in \mathcal{M}\}$
    • (ii) $\nu^-(E)= -\inf\{\nu(F)\colon F\subset E, F\in \mathcal{M}\}$
    • (iii) $|\nu|(E) = \sup\{ |\int_E f\ d\nu|\colon |f|\leq 1\}$
    • (iv) $|\nu|(E)=\sup\{|\nu(E_1)|+\cdots +|\nu(E_n)|\colon n\in \mathbb{N},\ E=E_1\cup\cdots\cup E_n\text{ is a}$ $\text{partition}\}$

Homework 6, due Wednesday, October 20th (Sections 2.3, 2.4) Solutions

• 1. Let $f\in L^1(\mathbb{R},m)$. Show that $F\colon \mathbb{R}\to\mathbb{C}$ is continuous where $F(t)=\int_{(-\infty,t]} f\ dm$.
• 2. Let $f\colon [a,b]\to \mathbb{R}$ be a bounded function and consider $h,H\colon [a,b]\to\mathbb{R}$ defined by $$h(t):= \lim_{\delta\to 0} \inf_{|s-t|\leq \delta} f(s) \qquad H(t):= \lim_{\delta\to 0} \sup_{|s-t|\leq \delta} f(s).$$
  • (a) Show that $f$ is continuous at $t\in [a,b]$ if and only if $h(t)=H(t)$.
  • (b) Show that $\int_{[a,b]} h\ dm$ and $\int_{[a,b]} H\ dm$ equal the lower and upper Darboux integrals of $f$, respectively.
    [Hint: show that $h=g$ and $H=G$ $m$-almost everywhere, where $g$ and $G$ are as in the proof of the Riemann–Lebesgue theorem.]
  • (c) Deduce that $f$ is Riemann integrable if and only if $$m(\{t\in [a,b]\colon f\text{ is discontinuous at }t\})=0.$$
• 3. Let $\{q_n\colon n\in\mathbb{N}\} =\mathbb{Q}$ be an enumeration of the rationals, and for $x\in \mathbb{R}$ define $$g(x):= \sum_{n=1}^\infty \frac{1}{2^n \sqrt{x-q_n}} 1_{(q_n,q_n+1)}$$
  • (a) Show that $g\in L^1(\mathbb{R},m)$ and hence $g<\infty$ $m$-almost everywhere.
  • (b) Show that $g$ is discontinuous everywhere and unbounded on every open interval.
  • (c) Show that the conclusions of (b) hold for any function equal to $g$ $m$-almost everywhere.
  • (d) Show that $g^2<\infty$ $m$-almost everywhere, but $g^2$ is not integrable on any interval.
• 4. Let $(X,\mathcal{M},\mu)$ be a measure space with $\mu(X)<\infty$. For $f,g\colon X\to \mathbb{C}$ $\mathcal{M}$-measurable define $$\rho(f,g) = \int_X \frac{|f-g|}{1+|f-g|}\ d\mu.$$
  • (a) Show that $\rho$ defines a metric on equivalence classes of $\mathbb{C}$-valued $\mathcal{M}$-measurable functions under the relation of $\mu$-almost everywhere equality.
  • (b) Show that $f_n\to f$ in measure if and only if $\rho(f_n,f)\to 0$.
• 5. (Lusin's Theorem) Let $f\colon [a,b]\to \mathbb{C}$ be Lebesgue measurable. Show that for all $\epsilon>0$ there exists a compact set $K\subset [a,b]$ with $m(K)>(b-a)-\epsilon$ such that $f|_K$ is continuous.
  [Hint: use Egoroff's theorem and the $L^1$-density of continuous functions.]

Exercises 1, 3, 4, and 5 were graded.

Homework 5, due Wednesday, October 13th (Sections 2.2, 2.3) Solutions

• 1. Let $f\colon [0,1]\to [0,1]$ be the Cantor function, and define $g(x):=f(x)+x$.
  • (a) Show that $g\colon [0,1]\to [0,2]$ is a bijection with continuous inverse.
  • (b) If $C\subset [0,1]$ is the Cantor set, show that $m(g(C))=1$. [Hint: compute $m(g(C)^c)$.]
  • (c) Show that there exists $A\subset g(C)$ such that $A\not\in \mathcal{L}$ and $g^{-1}(A)\in \mathcal{L}\setminus \mathcal{B}_\mathbb{R}$.
  • (d) Deduce that there exists a Lebesgue measurable function $F$ and a continuous function $G$ such that $F\circ G$ is not Lebesgue measurable.
• 2. Let $f\in L^+(X,\mathcal{M},\mu)$ with $\int_X f\ d\mu<\infty$.
  • (a) Show that $\{x\in X\colon f(x)=\infty\}$ is a $\mu$-null set.
  • (b) Show that $\{x\in X\colon f(x)>0\}$ is $\sigma$-finite.
  • (c) Show that for all $\epsilon>0$, there exists $E\in \mathcal{M}$ with $\mu(E)<\infty$ and such that $\int_X f\ d\mu < \int_E f\ d\mu + \epsilon$.
  • (d) Suppose $(f_n)_{n\in \mathbb{N}}\subset L^+(X,\mu)$ decreases to $f$ and $\int_X f_1\ d\mu <\infty$. Show that $$\lim_{n\to\infty} \int_X f_n\ d\mu = \int_X f\ d\mu.$$
• 3. For $f\in L^+(X,\mathcal{M},\mu)$, define $\nu\colon \mathcal{M}\to [0,\infty]$ by $\nu(E):= \int_E f\ d\mu$. Show that $\nu$ is a measure satisfying $$\int_X g\ d\nu = \int_X gf\ d\mu$$ for all $g\in L^+(X,\mathcal{M},\mu)$.
• 4. Let $(f_n)_{n\in \mathbb{N}}, (g_n)_{n\in \mathbb{N}} \in L^1(X,\mu)$ be sequences converging $\mu$-almost everywhere to $f,g\in L^1(X,\mu)$, respectively. Suppose $|f_n|\leq g_n$ for each $n\in \mathbb{N}$ and $\int g_n\ d\mu\to \int g\ d\mu$. Show that $$\lim_{n\to\infty} \int_X f_n\ d\mu = \int_X f\ d\mu.$$
• 5. Suppose $(f_n)_{n\in \mathbb{N}} \subset L^1(X,\mu)$ converges $\mu$-almost everywhere to $f\in L^1(X,\mu)$. Show that $$\lim_{n\to\infty} \int_X |f_n - f|\ d\mu=0 \qquad\Longleftrightarrow\qquad \lim_{n\to\infty} \int_X |f_n|\ d\mu = \int_X |f|\ d\mu.$$

Exercises 2-5 were graded.

Homework 4, due Wednesday, September 29th (Sections 1.5, 2.1) Solutions

• 1. Let $\mu$ be a Lebesgue--Stieltjes measure with domain $\mathcal{M}$, and let $E\in \mathcal{M}$ with $\mu(E)<\infty$. Show that for any $\epsilon>0$ there exists a finite union of open intervals $A$ so that $\mu(E\Delta A)<\epsilon$.
• 2. Let $N\subset [0,1)$ be the non-measurable set constructed in Section 1.1 and denote $$N_q:= \{x+q\colon x\in N\cap [0,1-q)\} \cup \{x+q-1\colon x\in N\cap [1-q,1)\},$$ for all $q\in \mathbb{Q}\cap [0,1)$. Let $E\subset \mathbb{R}$ be Lebesgue measurable.
  • (a) Show that $E\subset N$ implies $m(E)=0$.
  • (b) Show that $m(E)>0$ implies $E$ contains a subset that is not Lebesgue measurable. [Hint: for $E'\subset [0,1)$ one has $E'=\bigcup_q E'\cap N_q$.]
• 3. Let $E\subset\mathbb{R}$ be Lebesgue measurable with $m(E)>0$.
  • (a) Show that for any $0<\alpha<1$ there exists an interval $I$ satisfying $m(E\cap I) > \alpha m(I)$.
  • (b) Show that the set $$E-E:=\{x-y\colon x,y\in E\}$$ contains an open interval centered at $0$.
• 4. Let $(X,\mathcal{M})$ be a measurable space and $f,g\colon X\to\overline{\mathbb{R}}$.
  • (a) Show that $f$ is $\mathcal{M}$-measurable if and only if $f^{-1}(\{\infty\})$, $f^{-1}(\{-\infty\})$, $f^{-1}(B)\in \mathcal{M}$ for all Borel sets $B\subset \mathbb{R}$.
  • (b) Show that $f$ is $\mathcal{M}$-measurable if and only if $f^{-1}((q,\infty])\in \mathcal{M}$ for all $q\in \mathbb{Q}$.
  • (c) Suppose $f,g$ are $\mathcal{M}$-measurable. Fix $a\in \overline{\mathbb{R}}$ and define $h\colon X\to \overline{\mathbb{R}}$ by $$h(x) = \begin{cases} a & \text{if }f(x)=-g(x)=\pm \infty \\ f(x)+g(x) & \text{otherwise} \end{cases}.$$ Show that $h$ is $\mathcal{M}$-measurable.
• 5. Let $(X,\mathcal{M})$ be a measurable space, and let $f_n\colon X\to \overline{\mathbb{R}}$ be $\mathcal{M}$-measurable for each $n\in \mathbb{N}$. Show that $\{x\in X\colon {\displaystyle \lim_{n\to\infty} } f_n(x) \text{ exists}\}\in \mathcal{M}$.

Exercises 1, 2, 3, and 5 were graded, and 4 was checked for completion.

Homework 3, due Wednesday, September 22nd (Sections 1.4-1.5) Solutions

• 1. Let $\mathcal{A}$ be an algebra on $X$, let $\mu_0$ be a premeasure on $(X,\mathcal{A})$, and let $\mu^*$ be the outer measure defined by $$\mu^*(E):= \inf \left\{ \sum_{n=1}^\infty \mu_0(A_n)\colon A_n\in \mathcal{A}\text{ for each $n\in\mathbb{N}$ and }E\subset \bigcup_{n=1}^\infty A_n\right\}.$$ We will call $\mu^*$ the outer measure induced by $\mu_0$.
  • (a) Let $A_\sigma$ denote the collection of all countable unions of sets in $\mathcal{A}$. For $E\subset X$ and $\epsilon>0$ show that there exists $A\in \mathcal{A}_\sigma$ satisfying $E\subset A$ and $\mu^*(A)\leq \mu^*(E)+\epsilon$.
  • (b) Let $A_{\sigma\delta}$ denote the collection of all countable intersections of sets in $\mathcal{A}_\sigma$. For $E\subset X$ with $\mu^*(E)<\infty$, show that $E$ is $\mu^*$-measurable if and only if there exists $B\in \mathcal{A}_{\sigma\delta}$ satisfying $E\subset B$ and $\mu^*(B\setminus E)=0$.
  • (c) Suppose $\mu_0$ is $\sigma$-finite. Show that $E$ is $\mu^*$-measurable if and only if there exists $B\in A_{\sigma\delta}$ satisfying $E\subset B$ and $\mu^*(B\setminus E)=0$.
• 2. Let $\mathcal{A}$ be an algebra on $X$, let $\mu_0$ be a finite premeasure on $(X,\mathcal{A})$, and let $\mu^*$ be the outer measure induced by $\mu_0$. The inner measure induced by $\mu_0$ is the map defined by $\mu_*(E):=\mu_0(X) - \mu^*(E^c)$ for $E\subset X$. Show that $A\subset X$ is $\mu^*$-measurable if and only if $\mu^*(A)=\mu_*(A)$. [Hint: use Exercise 1.(b).]
• 3. Let $(X,\mathcal{M},\mu)$ be a $\sigma$-finite measure space and let $\mu^*$ be the outer measure induced by $\mu$.
  • (a) Show that the $\sigma$-algebra $\mathcal{M}^*$ of $\mu^*$-measurable sets equals $\overline{\mathcal{M}}$, the completion of $\mathcal{M}$. [Hint: use Exercise 1.(c).]
  • (b) Show that $\mu^*|_{\mathcal{M}^*} = \overline{\mu}$, the completion of $\mu$.
• 4. Let $\mathcal{A}$ be the collection of finite disjoint unions of sets of the form $(a,b]\cap \mathbb{Q}$ with $-\infty\leq a< b\leq \infty$.
  • (a) Show $\mathcal{A}$ is an algebra on $\mathbb{Q}$ by showing $\mathcal{E}:=\{\emptyset\}\cup\{ (a,b]\cap \mathbb{Q}\colon -\infty\leq a < b\leq b\}$ is an elementary family.
  • (b) Show that $\mathcal{M}(\mathcal{A}) = \mathcal{P}(\mathbb{Q})$.
  • (c) Define $\mu_0$ on $\mathcal{A}$ by $\mu_0(\emptyset)=0$ and $\mu_0(A)=\infty$ for all nonempty $A\in \mathcal{A}$. Show that $\mu_0$ is a premeasure.
  • (d) Show that there exists more than one measure on $(\mathbb{Q},\mathcal{P}(\mathbb{Q}))$ extending $\mu_0$.
• 5. Let $F\colon \mathbb{R}\to \mathbb{R}$ be increasing and right-continuous, and let $\mu_F$ be the associated Borel measure on $\mathbb{R}$. For $-\infty < a < b < \infty$, prove the following equalities: $$\begin{align*} \mu_F(\{a\}) &= F(a) - \lim_{x\nearrow a} F(x)\\ \mu_F([a,b]) &= F(b) - \lim_{x\nearrow a} F(x)\\ \mu_F((a,b)) &= \lim_{x\nearrow b} F(x) - F(a)\\ \mu_F([a,b)) &= \lim_{x\nearrow b} F(x) - \lim_{y\nearrow a} F(y). \end{align*}$$

Exercises 1,3, and 4 were graded, 2 and 5 were checked for completion.

Homework 2, due Wednesday, September 15th (Sections 1.2-1.3) Solutions

• 1. Fix $n\in \mathbb{N}$ and denote the Borel $\sigma$-algebra on $\mathbb{R}^n$ by $\mathcal{B}$.
  • (a) Show that $\mathcal{B}$ is generated by the collection of open boxes $$(a_1,b_1)\times \cdots \times (a_n,b_n)$$ for $a_1,b_1,\ldots, a_n,b_n\in \mathbb{R}$ with $a_1 < b_1,\ldots a_n < b_n$.
  • (b) Fix $\mathbf{t}=(t_1,\ldots, t_n)\in \mathbb{R}^n$. For a Borel set $E\subset \mathbb{R}^n$, show that $$E+\mathbf{t}:= \{ (x_1+t_1,\ldots, x_n+t_n)\colon (x_1,\ldots, x_n)\in E\}$$ is also a Borel set. We say $\mathcal{B}$ is translation invariant. [Hint: Consider the collection $\{E\in \mathcal{B}\colon E+\mathbf{t}\in \mathcal{B}\}$.]
  • (c) Fix $\mathbf{t}=(t_1,\ldots, t_n)\in \mathbb{R}^n$. For a Borel set $E\subset \mathbb{R}^n$, show that $$E\cdot \mathbf{t}:=\{ (x_1t_1,\ldots, x_nt_n)\colon (x_1,\ldots, x_n)\in E\}$$ is also a Borel set. We say $\mathcal{B}$ is dilation invariant.
• 2. Let $(X,\mathcal{M},\mu)$ be a measure space with $E_n\in \mathcal{M}$ for each $n\in \mathbb{N}$.
  • (a) Show that $$\mu(\liminf E_n) \leq \liminf_{n\to\infty} \mu(E_n).$$
  • (b) Suppose $\mu(\bigcup_n E_n)<\infty$. Show that $$\mu(\limsup E_n) \geq \limsup_{n\to\infty} \mu(E_n).$$
• 3. Let $\mu$ be a finitely additive measure on a measurable space $(X,\mathcal{M})$.
  • (a) Show that $\mu$ is a measure if and only if it satisfies continuity from below.
  • (b) If $\mu(X)<\infty$, show that $\mu$ is a measure if and only if it satisfies continuity from above.
• 4. Let $(X,\mathcal{M},\mu)$ be a measure space.
  • (a) Suppose $\mu$ is $\sigma$-finite. Show that $\mu$ is semifinite.
  • (b) Suppose $\mu$ is semifinite. Show that for $E\in \mathcal{M}$ with $\mu(E)=\infty$ and any $C > 0$, there exists $F\subset E$ with $C < \mu(F) < \infty$.
• 5. Let $(X,\mathcal{M},\mu)$ be a measure space and for $E\in \mathcal{M}$ define $$\mu_0(E):= \sup\{ \mu(F)\colon F\subset E \text{ with } \mu(F)<\infty\}.$$ We call $\mu_0$ the seminfinite part of $\mu$.
  • (a) Show that $\mu_0$ is a semifinite measure.
  • (b) Show that if $\mu$ is itself semifinite, then $\mu=\mu_0$.
  • (c) [Not Collected] We say $E\in \mathcal{M}$ is $\mu$-semifinite if for any $F\subset E$ with $\mu(F)=\infty$ there exists $G\subset F$ with $0 < \mu(G) < \infty$. Show that $$\nu(E):=\begin{cases} 0 & \text{if $E$ is $\mu$-semifinite} \\ \infty & \text{otherwise} \end{cases}$$ defines a measure on $(X,\mathcal{M})$ satisfying $\mu=\mu_0+\nu$.

Exercises 1,2,3, and 5 were graded, and 4 was checked for completion.

Homework 1, due Wednesday, September 8th (Sections 1.1-1.2) Solutions

• 1. Consider $f\colon \mathbb{R}\to \mathbb{R}$ defined by $$f(t) = \begin{cases} \frac{1}{n} & \text{if }t=\frac{m}{n}\text{ with $m\in \mathbb{Z}$, $n\in \mathbb{N}$ sharing no common factors}\\ 0 & \text{if }t\in \mathbb{R}\setminus\mathbb{Q}\end{cases}.$$
  • (a) Show that $f$ is discontinuous at every $t\in \mathbb{Q}$.
  • (b) Show that $f$ is continuous at every $t\in \mathbb{R}\setminus \mathbb{Q}$.
  • (c) Show that $1_\mathbb{Q}$ is discontinuous at every $t\in \mathbb{R}$.
• 2. Show that if $E\subset \mathbb{R}$ is countable then $E$ is a null set.
• 3. Let $X$ be a set and let $(E_n)_{n\in \mathbb{N}}$ be a sequence of subsets. Recall that the limit inferior and limit superior of this sequence of sets are defined as $$\liminf E_n := \bigcup_{n=1}^\infty \bigcap_{k=n}^\infty E_k \qquad\text{and}\qquad \limsup E_n:= \bigcap_{n=1}^\infty \bigcup_{k=n}^\infty E_k,$$ respectively. Show that for all $x\in X$, $$1_{\liminf E_n}(x) =\liminf_{n\to\infty} 1_{E_{n}}(x) \qquad \text{and}\qquad 1_{\limsup E_n}(x) = \limsup_{n\to\infty} 1_{E_n}(x).$$
• 4. Let $X$ be an uncountable set. Show that $$\mathcal{C}:=\{E\subset X\colon E \text{ or } E^c \text{ is countable}\}$$ is a $\sigma$-algebra on $X$.
• 5. Let $\mathcal{B}_\mathbb{R}$ be the Borel $\sigma$-algebra on $\mathbb{R}$, and consider the following collections of subsets of $\mathbb{R}$: $$\begin{align*} \mathcal{E}_1 &:=\{ (a,b)\colon a,b\in \mathbb{R},\ a < b\}\\ \mathcal{E}_2 &:=\{ [a,\infty) \colon a\in \mathbb{R}\}. \end{align*}$$ Show that $\mathcal{M}(\mathcal{E}_1)= \mathcal{M}(\mathcal{E}_2)=\mathcal{B}_\mathbb{R}$.

Exercises 1-5 were graded.

Midterm 1 is in class on Wednesday, October 6th. This covers Chapter 1 and Section 2.1 in the textbook. Solutions.

Midterm 2 is in class on Wednesday, November 17th. This covers Sections 2.2 - 2.5, and 3.1 - 3.3 in the textbook. Solutions.

The Final exam will be administered through D2L during a 3-hour window of you choosing between Wednesday, December 8th, 12:00 pm (noon) and Friday, December 17th, 11:59 pm. This covers Chapters 1-3 and Section 6.1 of the textbook.