REAL ANALYSIS GENERAL EXAM FALL 2022

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.

Compute $$\lim _{n \rightarrow \infty} \int_{0}^{\infty} \frac{n \sin (x / n)}{x\left(1+x^{2}\right)} d x$$

Problem 2.

Fix $a<b$ in $\mathbb{R}$. Recall that $h:[a, b] \rightarrow \mathbb{C}$ is absolutely continuous if for every $\varepsilon>0$ there is a $\delta>0$ so that if $\left(\left(a_{j}, b_{j}\right)\right)_{j=1}^{k}$ are disjoint intervals in $[a, b]$ with $\sum_{j=1}^{k}\left(b_{j}-\right. \left.a_{j}\right)<\delta$, then $\sum_{j=1}^{k}\left(f\left(b_{j}\right)-f\left(a_{j}\right)\right)<\varepsilon$. For a Lipschitz function $g:[a, b] \rightarrow \mathbb{C}$ we set $$\|g\|_{L i p}=\sup _{x \neq y, x, y \in[a, b]} \frac{|g(x)-g(y)|}{|x-y|}$$ (a) Show that $f:[a, b] \rightarrow \mathbb{C}$ is Lipschitz if and only if $f$ is absolutely continuous and $f^{\prime} \in L^{\infty}([a, b])$. (b) If $f:[a, b] \rightarrow \mathbb{C}$ is Lipschitz, show that $\|f\|_{\text {Lip }}=\left\|f^{\prime}\right\|_{\infty}$.

Problem 3.

Let $(X, \mu)$ be a $\sigma$-finite measure space. Show that if $f, g \in L^{1}(X, \mu)$ win $0 \leq f, g$ a.e., then $$\|f-g\|_{1}=\int_{0}^{\infty} \mu(\{x: f(x)>t\} \Delta\{x: g(x)>t\}) d t$$

Here $E \Delta F=E \backslash F \cup F \backslash E$ for sets $E, F \subseteq X$. Suggestion: it might be helpful to first show that for $a, b \in[0, \infty)$ we have $$|a-b|=\int_{0}^{\infty}\left|1_{(t, \infty)}(a)-1_{(t, \infty)}(b)\right| d t$$

Note: for this problem you may take for granted that the function $X \times(0, \infty) \rightarrow \{0,1\}$ given by $(y, t) \mapsto 1_{\{x: f(x)>t\}}(y)$ and that the function $t \mapsto \mu(\{x: f(x)> t\} \Delta\{x: g(x)>t\}$ ) are measurable functions.

Problem 4.

Let $(X, \Sigma)$ be a measurable space. Recall that if $\eta$ is a signed measure on $\Sigma$, then $|\eta|=\eta_{1}+\eta_{2}$ where $\eta_{1}, \eta_{2}$ are the unique nonnegative measures with $\eta=\eta_{1}-\eta_{2}$ and $\eta_{1} \perp \eta_{2}$. Further, $\|\eta\|_{T V}=|\eta|(X)$. Suppose that $\mu, \nu$ are signed measures on $\Sigma$, that $\|\mu\|_{T V},\|\nu\|_{T V}<+\infty$ and that $|\mu|,|\nu|$ are mutually singular. (a) If $\mu=\mu_{1}-\mu_{2}, \nu=\nu_{1}-\nu_{2}$ with $\mu_{i}, \nu_{j}$ nonnegative measures and $\mu_{1} \perp \mu_{2}$, $\nu_{1} \perp \nu_{2}$, show that $\mu_{i} \perp \nu_{j}$ for all $i, j \in\{1,2\}$. (b) Show that $$\|\mu+\nu\|_{T V}=\|\mu\|_{T V}+\|\nu\|_{T V}$$

Problem 5.

(a) For $f \in L^{1}([0,1])$, set $L_{f}$ be the set of $x \in[0,1]$ so that $$\lim _{r \rightarrow 0} \frac{1}{2 r} \int_{(x-r, x+r)}|f(y)-f(x)| d y=0$$

State the conclusion of the Lebesgue’s differentiation theorem for $L_{f}$. (b) For $n \in \mathbb{N}$, and $0 \leq j \leq 2^{n}-1$, set $I_{n, j}=\left[j 2^{-n},(j+1) 2^{-n}\right)$. For $f \in L^{1}([0,1])$, define $$E_{n} f=\sum_{j=0}^{2^{n}-1}\left(\frac{1}{m\left(I_{n, j}\right)} \int_{I_{n, j}} f(t) d t\right) 1_{I_{n, j}}$$

Show that $$\lim _{n \rightarrow \infty}\left(E_{n} f\right)(x)=f(x) \text { for almost every } x \in[0,1]$$