Real analysis general exam, January 2020

Problem 1

Let $\mu$ be the Lebesgue measure on $\mathbb{R}$. For a Lebesgue measurable set $A \subset[0,1]$, is it true that
(a) $\mu(A)=\sup _{U \subset A, U \text { open }} \mu(U)$ ? If true, prove this. If false, give a counterexample.
(b) $\mu(A)=\inf _{U \supset A, U \text { open }} \mu(U)$ ? If true, prove this. If false, give a counterexample.

Problem 2

Find a polynomial $P(x)$ of degree at most 3 such that $\int_{-1}^{1}\left|x^{4}-P(x)\right|^{2} d x$ is minimal.

Problem 3

Let $X$ be a compact metric space, and $C(X)$ be the space of all real-valued continuous functions on $X$ with the supremum norm. Assume that the subset $\mathcal{A} \subset C(X)$ satisfies the following properties:

• (algebra) For all $f, g \in \mathcal{A}$ and $\alpha, \beta \in \mathbb{R}$ we have $\alpha f+\beta g \in \mathcal{A}$ and $f g \in \mathcal{A}$.

• (separates points) For any $x \neq y$ from $X$ there exists a function $f \in \mathcal{A}$ such that $f(x) \neq f(y)$.

This question has two parts:
(a) Show by example that $\mathcal{A}$ need not be dense in $C(X)$, explicitly checking all the properties of your example $\mathcal{A}$.
(b) In order to conclude that $\mathcal{A}$ is dense by Stone-Weierstrass Theorem, what additional condition(s) should be added?

Problem 4

Let $\mu$ be a measure on ( $\mathbb{R}, \mathcal{B}$ ), where $\mathcal{B}$ is the Borel $\sigma$-algebra. Let $\mu(\mathbb{R})=1$. Next, let $\mathcal{F} \subset \mathcal{B}$ be the sub- $\sigma$-algebra of symmetric Borel sets, that is, $\mathcal{F}$ generated by all intervals of the form ( $-a, a$ ) with $a>0$.
Let $f \in L^{1}(\mathbb{R}, \mathcal{B}, \mu)$. Find a function $g$ such that:
(a) $g \in L^{1}(\mathbb{R}, \mathcal{F}, \mu)$ (in particular, $g$ is $\mathcal{F}$-measurable).
(b) For all $E \in \mathcal{F}$ we have $\int_{E} g d \mu=\int_{E} f d \mu$.

Problem 5

Let $\mu$ be a finite measure on some measurable space ( $X, \mathcal{F}$ ).

Show that a sequence of $\mathcal{F}$-measurable functions $f_{n}$ converges to a function $f$ in measure if and only if $\int_{X} \min \left\{1,\left|f_{n}-f\right|\right\} \mu(d x) \rightarrow 0$ as $n \rightarrow+\infty$.