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.