Real analysis Qualifying exam, August 2019

Problem 1

Let $\mathcal{C}$ be the Cantor set on $[0,1]$. Recall that it is obtained by iteratively deleting the open middle third: $\left(\frac{1}{3}, \frac{2}{3}\right)$, then $\left(\frac{1}{9}, \frac{2}{9}\right) \cup\left(\frac{7}{9}, \frac{8}{9}\right)$, and so on.
(a) Show that $\mathcal{C}+\mathcal{C}:=\{a+b: a, b \in \mathcal{C}\}$ is the full segment $[0,2]$.
(b) Find two sets $A, B \subset \mathbb{R}$, each of which is closed and has Lebesgue measure zero, such that $A+B=\{a+b: a \in A, b \in B\}$ is the full line $\mathbb{R}$.

Problem 2

Does there exist a measure space ( $X, \mathcal{F}, \mu$ ) with a finite measure $\mu$, and a sequence of $\mu$-measurable functions $\left\{f_{n}\right\}_{n=1,2, \ldots}$ on $X$ such that:

• $f_{n}(x) \geq 0$ for all $n, x$;

• $f_{n}(x) \rightarrow 0$ as $n \rightarrow+\infty$ for all $x$;

• $\int f_{n}(x) \mu(d x) \rightarrow 0$ as $n \rightarrow+\infty$;

• $\Phi(x):=\sup _{n} f_{n}(x)$ has infinite integral?

If yes, give an example of such a sequence $\left\{f_{n}\right\}$. If no, give a proof of nonexistence.

Problem 3

Let $\mu$ be a signed Borel measure on $\mathbb{R}^{n}$ which is bounded on bounded sets. Suppose that $\int f d \mu=0$ for all continuous functions $f$ with bounded support. Show that then $\mu=0$.

Problem 4

Let $L^{1}(\mathbb{R})$ be the space of Lebesgue integrable functions on $\mathbb{R}$. For a positive function $f \in L^{1}(\mathbb{R})$ show that the function $\frac{1}{f(x)}$ does not belong to $L^{1}(\mathbb{R})$.
(Hint: look at the function $1=f^{1 / 2} f^{-1 / 2}$.)

Problem 5

Applying the Gram-Schmidt orthogonalization to $1, x, x^{2}, \ldots$ in the Hilbert space $L^{2}([-1,1])$ (with Lebesgue measure), one gets the Legendre polynomials $L_{n}(x), n=0,1,2, \ldots$.
(a) Show that the Legendre polynomials form a basis (= complete orthogonal system) in the Hilbert space $L^{2}([-1,1])$
(b) Show that the Legendre polynomials are given by the formula $L_{n}(x)=c_{n} \frac{d^{n}}{d x^{n}}\left(x^{2}-1\right)^{n}$ (you do not need to specify $c_{n}$ ).
(Hint: employ integration by parts.)