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*}

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.

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*}

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*}