Fall 2018 Practice Midterm 1.1 #real_analysis/qual/work
Let m∗(E) denote the Lebesgue outer measure of a set E⊆Rn.
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Prove using the definition of Lebesgue outer measure that m(∞⋃j=1Ej)≤∞∑j=1m∗(Ej).
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Prove that for any E⊆Rn and any ϵ>0 there exists an open set G with E⊆G and m∗(E)≤m∗(G)≤m∗(E)+ϵ.
Fall 2018 Practice Midterm 1.2 #real_analysis/qual/work
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See
\cref{equivalence_of_approximating_measures} -
Let fk be a sequence of extended real-valued Lebesgue measurable function.
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Prove that inf are both Lebesgue measurable function.
Hint: argue that \begin{align*} \left\{{x {~\mathrel{\Big\vert}~}\inf_k f_k(x) < a}\right\} = \bigcup_k \left\{{x {~\mathrel{\Big\vert}~}f_k(x) < a}\right\} .\end{align*}
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Carefully state Fatou’s Lemma and deduce the Monotone Converge Theorem from it.
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Fall 2018 Practice Midterm 1.3 #real_analysis/qual/work
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Prove that if f, g\in L^+({\mathbf{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^+({\mathbf{R}}^n) then \begin{align*} \int \sum_k f_k = \sum_k \int f_k .\end{align*}
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Let \left\{{E_j}\right\}_{j\in {\mathbb{N}}} \subseteq \mathcal{M}({\mathbf{R}}^n) with E_j \nearrow E. Use the countable additivity of \mu_f on \mathcal{M}({\mathbf{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
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Show that f\in L^1({\mathbf{R}}^n) \implies {\left\lvert {f(x)} \right\rvert} < \infty almost everywhere.
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Show that if \left\{{f_k}\right\} \subseteq L^1({\mathbf{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*}