Practice Exam (November 2014)

Fall 2018 Practice Midterm 1.1 #real_analysis/qual/work

Let $$m_*(E)$$ denote the Lebesgue outer measure of a set $$E \subseteq {\mathbb{R}}^n$$.

• Prove using the definition of Lebesgue outer measure that \begin{align*} m \qty{ \displaystyle\bigcup_{j=1}^{\infty } E_j } \leq \sum_{j=1}^{\infty } m_*(E_j) .\end{align*}

• Prove that for any $$E \subseteq {\mathbb{R}}^n$$ and any $$\epsilon> 0$$ there exists an open set $$G$$ with $$E \subseteq G$$ and \begin{align*} m_*(E) \leq m_*(G) \leq m_*(E) + \epsilon .\end{align*}

Fall 2018 Practice Midterm 1.2 #real_analysis/qual/work

• See

\cref{equivalence_of_approximating_measures}

• Let $$f_k$$ be a sequence of extended real-valued Lebesgue measurable function.

• Prove that $$\inf_k f_k, \sup_k f_k$$ are both Lebesgue measurable function.

Hint: argue that \begin{align*} \left\{{x {~\mathrel{\Big\vert}~}\inf_k f_k(x) < a}\right\} = \displaystyle\bigcup_k \left\{{x {~\mathrel{\Big\vert}~}f_k(x) < a}\right\} .\end{align*}

• Carefully state Fatou’s Lemma and deduce the Monotone Converge Theorem from it.

Fall 2018 Practice Midterm 1.3 #real_analysis/qual/work

• Prove that if $$f, g\in L^+({\mathbb{R}})$$ then \begin{align*} \int(f +g) = \int f + \int g .\end{align*} Extend this to establish that if $$\left\{{ f_k}\right\} \subseteq L^+({\mathbb{R}}^n)$$ then \begin{align*} \int \sum_k f_k = \sum_k \int f_k .\end{align*}

• Let $$\left\{{E_j}\right\}_{j\in {\mathbb{N}}} \subseteq \mathcal{M}({\mathbb{R}}^n)$$ with $$E_j \nearrow E$$. Use the countable additivity of $$\mu_f$$ on $$\mathcal{M}({\mathbb{R}}^n)$$ established above to show that \begin{align*} \mu_f(E) = \lim_{j\to \infty } \mu_f(E_j) .\end{align*}

Fall 2018 Practice Midterm 1.4 #real_analysis/qual/work

• Show that $$f\in L^1({\mathbb{R}}^n) \implies {\left\lvert {f(x)} \right\rvert} < \infty$$ almost everywhere.

• Show that if $$\left\{{f_k}\right\} \subseteq L^1({\mathbb{R}}^n)$$ with $$\sum {\left\lVert {f_k} \right\rVert}_1 < \infty$$ then $$\sum f_k$$ converges almost everywhere and in $$L^1$$.

• Use the Dominated Convergence Theorem to evaluate \begin{align*} \lim_{t\to 0} \int_0^1 {e^{tx^2} - 1 \over t} \,dx .\end{align*}
#real_analysis/qual/work