Midterm Exam 1 (October 2018)

Fall 2018 Midterm 1.1 #real_analysis/qual/work

\label{equivalence_of_approximating_measures}

Let

$E \subseteq {\mathbf{R}}^n$ be bounded. Prove the following are equivalent:

For any $\epsilon>0$ there exists and open set $G$ and a closed set $F$ such that $\begin{align*} F \subseteq E \subseteq G && m(G\setminus F) < \epsilon .\end{align*}$
There exists a $G_ \delta$ set $V$ and an $F_ \sigma$ set $H$ such that $\begin{align*} m(V\setminus H) = 0 .\end{align*}$

Let $\left\{{ f_k }\right\} _{k=1}^{\infty }$ be a sequence of extended real-valued Lebesgue measurable functions.

Prove that $\sup_k f_k$ is a Lebesgue measurable function.
Prove that if $\lim_{k \to \infty } f_k(x)$ exists for every $x \in {\mathbf{R}}^n$ then $\lim_{k\to \infty } f_k$ is also a measurable function.

Prove that if $E \subseteq {\mathbf{R}}^n$ is a Lebesgue measurable set, then for any $h \in {\mathbf{R}}$ the set $\begin{align*} E+h \coloneqq\left\{{x + h {~\mathrel{\Big\vert}~}x\in E }\right\} \end{align*}$ is also Lebesgue measurable and satisfies $m(E + h) = m(E)$ .
Prove that if $f$ is a non-negative measurable function on ${\mathbf{R}}^n$ and $h\in {\mathbf{R}}^n$ then the function $\begin{align*} \tau_h d(x) \coloneqq f(x-h) \end{align*}$ is a non-negative measurable function and $\begin{align*} \int f(x) \,dx= \int f(x-h) \,dx .\end{align*}$

Let $f: {\mathbf{R}}^n\to {\mathbf{R}}$ be a Lebesgue measurable function.

Prove that for all $\alpha> 0$ , $\begin{align*} A_ \alpha \coloneqq\left\{{x\in {\mathbf{R}}^n {~\mathrel{\Big\vert}~}{\left\lvert { f(x) } \right\rvert} > \alpha}\right\} \implies m(A_ \alpha) \leq {1\over \alpha} \int {\left\lvert {f (x)} \right\rvert} \,dx .\end{align*}$
Prove that $\begin{align*} \int {\left\lvert { f(x) } \right\rvert} \,dx= 0 \iff f = 0 \text{ almost everywhere} .\end{align*}$

Let $\left\{{ f_k }\right\}_{k=1}^{\infty } \subseteq L^2([0, 1])$ be a sequence which converges in $L^1$ to a function $f$ .

Prove that $f\in L^1([0, 1])$ .
Give an example illustrating that $f_k$ may not converge to $f$ almost everywhere.

Prove that $\left\{{f_k}\right\}$ must contain a subsequence that converges to $f$ almost everywhere.