Homework is collected at the beginning of lecture of the specified day. No late homework will be accepted. Please write your name and the homework number on each assignment. If you would like to try writing your homework with LaTeX, here is a template that produces this output.

Homework 10, due Wednesday, December 6th (Section 3.5) Solutions

• 1. 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 $G\not\in BV([-1,1])$.
• 2. Suppose $(F_n)_{n\in \mathbb{N}}\subset BV$ converges pointwise to a function $F\in BV$. Show that $T_F\leq \displaystyle\liminf_{n\to\infty} T_{F_n}$.
• 3. For $F\in BV$ define \[ \|F\|_{BV}:=|F(0)| + T_F(\infty). \]
  • (a) Show that $\|\cdot\|_{BV}$ defines a norm on $BV$.
  • (b) Show that $BV$ is complete with respect to the metric $\|F - G\|_{BV}$.
  • (c) Show that for any $x_0\in \mathbb{R}$ one has \[ \frac12 \|F\|_{BV}\leq |F(x_0)| + T_F(\infty) \leq 2 \|F\|_{BV} \] for all $F\in BV$.
• 4. Suppose $F,G\colon [a,b]\to \mathbb{C}$ are absolutely continuous functions and that $G(x)\neq 0$ for all $x\in [a,b]$. Show that the quotient $\frac{F}{G}$ is absolutely continuous.
• 5. 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 $\int_{[a,b]} (f\circ G) G'\ dm$.
• 6*. 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 $t=g(x)$.]

Exercise 6 will not be collected.

Homework 9, due Wednesday, November 22nd (Sections 3.3, 3.4) Solutions

• 1. Let $\nu$ be a complex measure on $(X,\mathcal{M})$. Show that $\nu=|\nu|$ iff $\nu(X)=|\nu|(X)$.
• 2. 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*}
• 3. 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.]
• 4. 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 $|\nu|=|\lambda|+|\rho|$.]
• 5*. 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 $L^1(\mu)$.
• 6*. 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.]

Exercises 5 and 6 will not be collected.

Homework 8, due Wednesday, November 8th (Sections 3.1, 3.2) Solutions

• 1. 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) assuming $\nu$ is $\sigma$-finite, \[ |\nu|(E) = \sup\{ |\int_E f\ d\nu|\colon f\in L^1(X,\nu) \text{ and } |f|\leq 1\} \]
    • (iv) $|\nu|(E)=\sup\{|\nu(E_1)|+\cdots +|\nu(E_n)|\colon n\in \mathbb{N},\ E=E_1\cup\cdots\cup E_n\text{ is a}$ $\text{partition}\}$
• 2. Let $(\nu_n)_{n\in \mathbb{N}}$ be a sequence of positive measures and let $\mu$ be a positive measure, all defined on the same measurable space $(X,\mathcal{M})$. Denote $\nu = \sum_{n=1}^\infty \nu_n$.
  • (a) Show that if $\nu_n\perp \mu$ for all $n\in \mathbb{N}$, then $\nu \perp \mu$.
  • (b) Show that if $\nu_n \ll \mu$ for all $n\in \mathbb{N}$, then $\nu\ll \mu$.
• 3. 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 $(x_1,x_2)\in X_1\times X_2$.
• 4. 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.]
• 5. 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 \[ \frac{d\nu}{d\mu} = \frac{\frac{d \nu}{d\lambda}}{1-\frac{d \nu}{d\lambda}}. \]

Homework 7, due Wednesday, November 1st (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 $(X,\mathcal{M},\mu)$ and $(Y,\mathcal{N},\nu)$ be measure spaces, let $f\colon X\to \mathbb{C}$ be $\mathcal{M}$-measurable, and let $g\colon Y\to \mathbb{C}$ be $\mathcal{N}$-measurable.
  • (a) Show that $h\colon X\times Y\to\mathbb{C}$ defined by $h(x,y):=f(x)g(y)$ is $\mathcal{M}\otimes \mathcal{N}$-measurable.
  • (b) Show that if $f\in L^1(X,\mu)$ and $g\in L^1(Y,\nu)$, then $h\in L^1(X\times Y,\mu\times \nu)$ with \[ \int_{X\times Y} h \ d(\mu\times \nu) = \left(\int_X f\ d\mu\right) \left( \int_Y g\ d\nu\right). \]

Homework 6, due Wednesday, October 18th (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.]

Homework 5, due Wednesday, October 11th (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 \[ \lim_{n\to\infty} \int_X |f_n - f|\ d\mu=0 \qquad\Longleftrightarrow\qquad \lim_{n\to\infty} \int_X |f_n|\ d\mu = \int_X |f|\ d\mu. \]

Homework 4, due Wednesday, September 27th (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 $\{x\in X\colon {\displaystyle \lim_{n\to\infty} } f_n(x) \text{ exists}\}\in \mathcal{M}$.

Homework 3, due Wednesday, September 20th (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*}

Homework 2, due Wednesday, September 13th (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 $\mathbf{t}=(t_1,\ldots, t_n)\in (\mathbb{R}\setminus\{0\})^n$. 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 $\mu=\mu_0+\nu$.

Homework 1, due Wednesday, September 6th (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 3, 4, and 5 were graded, and the rest were checked for completion.

Midterm 1 is in class on Wednesday, October 4th. This covers Chapter 1 and Section 2.1 in the textbook. Solutions.

Midterm 2 is in class on Wednesday, November 15th. This covers Chapter 2 and Sections 3.1 and 3.2 in the textbook. Solutions.

The Final exam will be administered through D2L during a 3-hour window of you choosing between Monday, December 4th, 9:00 am and Friday, December 15th, 12:00 pm (noon).