REAL ANALYSIS GENERAL EXAM JANUARY 2025

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 ( $X, \mathcal{M}, \mu$ ) be a measure space and let ( $f_{n}$ ), ( $g_{n}$ ) be sequences of functions in $L^{1}(X, \mathcal{M}, \mu)$ that converge pointwise a.e. to functions $f, g \in L^{1}(X, \mathcal{M}, \mu)$ respectively. Suppose that $\left|f_{n}\right| \leq g_{n}$ a.e. and that

$$\lim _{n \rightarrow \infty} \int_{X} g_{n}(x) d \mu(x)=\int_{X} g(x) d \mu(x)$$

Show that

$$\lim _{n \rightarrow \infty} \int_{X} f_{n}(x) d \mu(x)=\int_{X} f(x) d \mu(x)$$

Problem 2

(a) Let $(X, \mathcal{M}, \mu)$ be a finite measure space. Show that if $p, p^{\prime} \in[1, \infty]$ with $p<p^{\prime}$, then $L^{p}(X, \mathcal{M}, \mu) \supseteq L^{p^{\prime}}(X, \mathcal{M}, \mu)$.
(b) Show that if $p, p^{\prime} \in[1, \infty]$ with $p<p^{\prime}$, the spaces $L^{p}(\mathbb{R}) \backslash L^{p^{\prime}}(\mathbb{R})$ and $L^{p^{\prime}}(\mathbb{R}) \backslash L^{p}(\mathbb{R})$ are both nonempty. (Note that the Lebesgue measure on $\mathbb{R}$ is not a finite measure, but merely a $\sigma$-finite measure.)

Problem 3

Let $\mathcal{H}$ be a separable Hilbert space. A sequence ( $v_{m}$ ) in $\mathcal{H}$ converges weakly to $v \in \mathcal{H}$ if

$$\lim _{m \rightarrow \infty}\left\langle v_{m}, w\right\rangle=\langle v, w\rangle$$

for every $w \in \mathcal{H}$. Show that for any sequence ( $v_{m}$ ) in $\mathcal{H}$ for which $\sup _{m \in \mathbb{N}}\left\|v_{m}\right\|$ is finite, there exists a subsequence ( $v_{m_{k}}$ ) that converges weakly to some $v \in \mathcal{H}$.

Problem 4

Define the Dirac delta measure $\delta_{0}$ on the Borel $\sigma$-algebra $\mathcal{B}_{\mathbb{R}}$ of $\mathbb{R}$ by

$$\delta_{0}(A):= \begin{cases}1 & \text { if } A \ni 0 \\ 0 & \text { otherwise }\end{cases}$$

For each $r>0$, let $\nu_{r}$ be the measure on the Borel $\sigma$-algebra $\mathcal{B}_{\mathbb{R}}$ of $\mathbb{R}$ given by

$$\nu_{r}(A):=\frac{1}{2 r} m(A \cap[-r, r]),$$

where $m$ denotes the Lebesgue measure on $\mathbb{R}$. Show that for every continuous function $f: \mathbb{R} \rightarrow \mathbb{C}$, we have that

$$\lim _{r \searrow 0} \int_{\mathbb{R}} f(x) d \nu_{r}(x)=\int_{\mathbb{R}} f(x) d \delta_{0}(x) .$$

Problem 5

(a) State the Riemann-Lebesgue lemma for the Fourier transform on $\mathbb{R}^{n}$.
(b) Show that there does not exist a function $g \in L^{1}\left(\mathbb{R}^{n}\right)$ that satisfies $f * g=f$ for all $f \in L^{1}\left(\mathbb{R}^{n}\right)$.