REAL ANALYSIS GENERAL EXAM SPRING 2023

Solve as many problems as you can. Full solutions on a smaller number of problems will be worth more than partial solutions on several problems.

Problem 1.

Let ( $X, \mu$ ) be a $\sigma$-finite measure space and $p \in[1,+\infty)$. Let $\left(f_{n}\right)_{n=1}^{\infty}$ be a sequence in $L^{p}(X, \mu)$ and suppose that $\left\|f_{n}\right\|_{p} \leq 1$, and that $f_{n}$ converges a.e. to a measurable function $f$. Show that $\|f\|_{p} \leq 1$.

Problem 2.

Let $\mu$ be a Borel probability measure on $\mathbb{R}$ without atoms. Suppose that $E \subseteq \mathbb{R}$ is a Borel set with $\mu(E)>0$. Show that there is a $t \in \mathbb{R}$ with $\mu(E \cap(-\infty, t))=\frac{1}{2} \mu(E)$.

Problem 3.

Let $X$ be a set equipped with a $\sigma$-algebra of sets $\Sigma$. Suppose that $\mu, \nu: \Sigma \rightarrow [0,+\infty)$ are finite measures. Set $\lambda=\mu+\nu$. Let $f: X \rightarrow \mathbb{R}$ be any $\Sigma$-measurable function so that

$$\nu(E)=\int_{E} f d \lambda$$

for all $E \in \Sigma$.
(i) Show that $0 \leq f \leq 1 \,\lambda$-a.e.
(ii) If $F=\{x: f(x)=1\}$, show that $\mu(F)=0$.
(iii) If $A \subseteq\{x: 0 \leq f(x)<1\}$ and $\mu(A)=0$, show that $\nu(A)=0$.

Problem 4.

Fix $p \in[1,+\infty)$. Let $W^{p}([0,1])$ consist of all absolutely continuous functions $f:[0,1] \rightarrow \mathbb{C}$ so that $f^{\prime} \in L^{p}([0,1])$. For $f \in W^{p}([0,1])$ define

$$\|f\|=|f(0)|+\left\|f^{\prime}\right\|_{p}$$

Show that $\|\cdot\|$ is a norm which makes $W^{p}([0,1])$ into a Banach space. (You are allowed to use that $L^{p}([0,1])$ is a Banach space).

Problem 5.

Let $m$ be Lebesgue measure on $\mathbb{R}$. Let $\Omega=\left\{1_{E}: E \subseteq \mathbb{R}\right.$ is Borel and $m(E)< +\infty\}$ regarded as a subset of $L^{1}(\mathbb{R})$ (recall that we identify two elements of $L^{1}$ if they agree almost everywhere). Throughout this problem regard $\Omega$ as a metric space equipped with the $L^{1}$-distance.
(i) If $a<b$ are real numbers, show that the function $\Omega \rightarrow \mathbb{R}$ given by

$$1_{E} \mapsto m(E \cap[a, b])$$

is a continuous function.
(ii) If $a<b$ are real numbers, let $U_{a, b}$ be the subset of $\Omega$ consiting of all $1_{E}$ where $E \subseteq \mathbb{R}$ is Borel and

$$0<m(E \cap[a, b])<b-a$$

Show that $U_{a, b}$ is open and dense in $\Omega$.
(iii) Let $D$ be the set of all $1_{E}$ where $E \subseteq \mathbb{R}$ is Borel and so that for every interval $I$ of positive measure we have

$$0<m(E \cap I)<m(I)$$

Show that there is a countable collection $\left\{U_{j}\right\}_{j \in J}$ of open and dense subsets of $\Omega$ with

$$D \supseteq \bigcap_{j \in J} U_{j} .$$