Real Analysis General Exam January 2022

Instructions. 2 hours. Closed book examination. Be neat in your presentation. When invoking a theorem from previous courses, name the theorem and thoroughly check its hypotheses. You must solve a significant portion of each of the three problems in order to pass the exam.

Problem 1

Let ( $X, \mathscr{M}, \mu$ ) be a measure space. Let ( $f_{n}$ ) be a sequence of nonnegative functions functions in $L^{1}(X, \mathscr{M}, \mu)$ and let $f$ be a nonnegative function in $L^{1}(X, \mathscr{M}, \mu)$. Suppose that

$$\int_{X} f_{n} \mathrm{~d} \mu \longrightarrow \int_{X} f \mathrm{~d} \mu$$

and that $f_{n} \rightarrow f$ pointwise. Prove that $f_{n}$ converges to $f$ in $L^{1}(X, \mathscr{M}, \mu)$. Hint: consider $g_{n}=\min \left(f, f_{n}\right)$.

Problem 2

Let ( $X, \mathscr{M}, \mu$ ) be a measure space. Let $p \in[1, \infty)$.
(a) Let ( $f_{n}$ ) be a sequence of functions in $L^{p}(X, \mathscr{M}, \mu)$ and let $f$ be a function in $L^{p}(X, \mathscr{M}, \mu)$. Suppose that $f_{n}$ converges to $f$ in $L^{p}(X, \mathscr{M}, \mu)$. Prove that there exists a subsequence ( $f_{n_{k}}$ ) such that for $\mu$-almost all $x, \lim _{k \rightarrow \infty} f_{n_{k}}(x)=f(x)$. Hint: remember the proof of completeness of $L^{p}$.
(b) Let $h$ be a measurable function on $X$. Let

$$D=\left\{f \in L^{p}(X, \mathscr{M}, \mu) \mid h f \in L^{p}(X, \mathscr{M}, \mu)\right\}$$

Let $\left(f_{n}\right)$ be a sequence of elements of $D$, and let $f, g \in L^{p}(X, \mathscr{M}, \mu)$ be such that $f_{n}$ converges to $f$ in $L^{p}$, and $h f_{n}$ converges to $g$ in $L^{p}$. Show that $f \in D$ and $g=h f$.

Problem 3

For $\mu$ a Borel probability measure on $\mathbb{R}$, we will denote by $\widehat{\mu}$ the function $\mathbb{R} \rightarrow \mathbb{C}$ given by

$$\widehat{\mu}(t)=\int_{\mathbb{R}} e^{i t x} \mathrm{~d} \mu(x)$$

We will also adopt the notational convention $\operatorname{sinc}(x)=\frac{\sin x}{x}$ if $x \neq 0$ and $\operatorname{sinc}(0)=1$.
(a) Show that $\widehat{\mu}$ is a bounded continuous function.
(b) Let $\delta>0$. Show that

$$\frac{1}{2 \delta} \int_{-\delta}^{\delta}(1-\operatorname{Re}(\widehat{\mu}(t))) \mathrm{d} t=\int_{\mathbb{R}}(1-\operatorname{sinc}(\delta x)) \mathrm{d} \mu(x)$$

(c) Show that for all $u \in \mathbb{R}$,

$$1-\operatorname{sinc}(u) \geq \frac{1}{2} \chi_{(-\infty,-2) \cup(2, \infty)}(u)$$

and deduce that

$$\mu\left(\left\{x \in \mathbb{R}\left||x|>2 \delta^{-1}\right\}\right) \leq \frac{1}{\delta} \int_{-\delta}^{\delta}(1-\operatorname{Re}(\widehat{\mu}(t))) \mathrm{d} t\right.$$

(d) Let $\mu_{n}$ be a sequence of Borel probability measures on $\mathbb{R}$. Suppose that for all $t$, the limit $\Phi(t)=\lim _{n \rightarrow \infty} \widehat{\mu_{n}}(t)$ exists and that the resulting function $\Phi(t)$ is continuous at $t=0$. Prove that for all $\epsilon>0$, there exists a compact set $K$ inside $\mathbb{R}$ such that, for all $n, \mu_{n}(K) \geq 1-\epsilon$.