Midterm Exam 1 (October 2018)

Fall 2018 Midterm 1.1 #real_analysis/qual/work

\label{equivalence_of_approximating_measures}
Let $$E \subseteq {\mathbb{R}}^n$$ 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*}

Fall 2018 Midterm 1.2 #real_analysis/qual/work

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 $$x \in {\mathbb{R}}^n$$ then $$\lim_{k\to \infty } f_k$$ is also a measurable function.

Fall 2018 Midterm 1.3 #real_analysis/qual/work

• Prove that if $$E \subseteq {\mathbb{R}}^n$$ is a Lebesgue measurable set, then for any $$h \in {\mathbb{R}}$$ 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 $${\mathbb{R}}^n$$ and $$h\in {\mathbb{R}}^n$$ 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*}

Fall 2018 Midterm 1.4 #real_analysis/qual/work

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

• Prove that for all $$\alpha> 0$$ , \begin{align*} A_ \alpha \coloneqq\left\{{x\in {\mathbb{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*}

Fall 2018 Midterm 1.5 #real_analysis/qual/work

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.
#real_analysis/qual/work