## Real Analysis Exam

The real analysis part of a phd exam in mathematics ## 2004 April 19

Instructions. Use a separate sheet of paper for each new problem. Do as many problems as you can. Complete solutions to five problems will be considered as an excellent performance. Be advised that a few complete and well written solutions will count more than several partial solutions.
Notation. $f\in C(X)$ means that $f$ is a real-valued, continuous function defined on $X$.

1. (a) Let $S$ be a (Lebesgue) measurable subset of $\mathbb R$ and let $f, g: S\to \mathbb R$ be measurable functions. Prove:
i. $f+g$ is measurable and
ii. if $\phi \in C(\mathbb R)$, then $\phi(f)$ is measurable.
(b) Let $f\colon [a,b]\to [-\infty, \infty]$ be a measurable function. Suppose that $f$ takes the value $\pm \infty$ only on a set of (Lebesgue) measure zero. Prove that for each $\epsilon>0$ there is a positive number $M$ such that $|f|\leq M$, except on a set of measure less than $\epsilon$.

2. (a) State Egorov's theorem.
(b) State Fatou's lemma.
(c) Let $\{f_n\} \subset L^p[0,1]$, where $1\leq p< \infty$. Suppose that $f_n\to f$ a.e., where $f\in L^p[0,1]$. Prove that $\|f_n - f\|_p\to 0$ if and only if $\|f_n\|_p\to \|f\|_p$.

3. (a) Let $S = [0, 1]$ and let $\{f_n\} \subset L^p(S)$, where $1< p< \infty$. Suppose that $f_n\to f$ a.e. on $S$, where $f\in L^p(S)$. If there is a constant $M$ such that $|f_n|_p\leq M$ for all $n$, prove that for each $g\in L^q(S), \frac{1}{p} + \frac{1}{q} = 1$, we have $\lim\limits_{n\to \infty} \int_S f_n g = \int_S fg$.
(b) Show by means of an example that this result is false for $p=1$.

4. State and prove the closed graph theorem.

5. Prove or disprove:
(a) For $1 \leq p < \infty$, let $\ell^p = \left\{ \mathbf{x} = \{x_k\} \bigm\vert \| \mathbf x \|_p = \left(\sum_{k=1}^\infty|x_k|^p\right)^{1/p} < \infty \right\}$. Then for $p \neq 2$, $\ell^p$ is a Hilbert space.
(b) Let $X = (C[0,1], \| \cdot \|_1)$, where the linear space $C[0,1]$ is endowed with the $L^1$-norm: $\|f\|_1 = \int_0^1 |f(x)|\ dx$. Then $X$ is a Banach space.
(c) Every real, separable Hilbert space is isometrically isomorphic to $\ell^2$.

6. Let $f, g \in L^1(\mathbb R)$. Give a precise statement of some version of Fubini's theorem that is valid for non-negative functions, and then prove the following:
(a) $h(x) = \int_\mathbb R f(x-t)g(t)\ dt$ exists for almost all $x\in \mathbb R$;
(b) $h\in L^1(\mathbb R)$ and $\|h\|_1 \leq \|f\|_1 \|g\|_1$.

7. (a) State the Radon-Nikodym theorem.
(b) Let $(X, \mathcal B, \mu)$ be a complete measure space, where $\mu$ is a positive measure defined on $\mathcal B$, a $\sigma$-algebra of subsets of $X$. Suppose $\mu(X) < \infty$ and $S$ is a closed subset of $\mathbb R$. Let $f\in L^1(\mu)$, where $f$ is an extended real-valued function defined on $X$. Prove: If for every $E\in \mathcal B$ with $\mu(E) > 0$ we have $$A_E(f) = \frac{1}{\mu(E)}\int_E f\ d\mu \in S,$$ then $f(x)\in S$ for almost all $x\in X$.