Bressan problems

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 18 March 2014

Bressan problems

1. Let $\left\{A_i\hspace{0.17em}\right|\hspace{0.17em}i\in I\}$ be an open covering of a compact metric space $K\text{.}$ Prove that there exists $\rho \in {ℝ}_{>0}$ such that if $x\in K$ then there exists $j\in I$ such that $B(x,\rho )\subseteq A_j\text{.}$
2. Let $x_1,x_2,\dots$ be a sequence in a metric space $E\text{.}$ Prove the following.
   1. The sequence $x_1,x_2,\dots$ converges to $x$ if and only if from every subsequence ${\left(x_{n_j}\right)}_{j\in {\mathbb{Z}}_{>0}}$ one can extract a further subsequence converging to $x\text{.}$
   2. If $d(x_m,x_n)\ge \delta$ where $\delta \in {ℝ}_{>0}$ for all $m\ne n$ then no convergent subsequence can exist.
   3. Let $E$ be complete and assume that, for every $\epsilon >0,$ from any sequence once can extract a further subsequence ${\left(x_{n_j}\right)}_{j\in {\mathbb{Z}}_{>0}}$ such that $$\underset{j,k\to \infty }{\text{limsup}}\hspace{0.17em}d(x_{n_j},x_{n_k})<\epsilon \text{.}$$ Then the sequence admits a convergent subsequence.
3. Consider the function $$f\left(x\right)=\{\begin{array}{cc}\frac{1}{x{\left(\text{ln}\hspace{0.17em}x\right)}^2},& \text{if}\hspace{0.17em}0<x<\frac{1}{2},\\ 0,& \text{otherwise.}\end{array}$$ Let $f_{\epsilon }\doteq J_{\epsilon }*f$ be the corresponding mollifications, and let $$F\left(x\right)\doteq \underset{0<\epsilon <1}{\text{sup}}{f}_{\epsilon }\left(x\right)\text{.}$$ Prove that $f\in L^1\left(ℝ\right)$ but $F\notin L^1\left(ℝ\right)\text{.}$ As a consequence, although $f_{\epsilon }\to f$ pointwise, one cannot use the Lebesgue dominated convergence theorem to prove that ${\Vert f_{\epsilon }-f\Vert }_{L^1\left(ℝ\right)}\to 0\text{.}$
4. Let $f_n:ℝ\to ℝ$ for $n\in {\mathbb{Z}}_{\ge 1},$ be a sequence of absolutely continuous functions such that
   1. at the point $x=0,$ the sequence $f_n\left(0\right)$ is bounded,
   2. there exists a function $g\in L^1\left(ℝ\right)$ such that the derivatives $f_n^{\prime }$ satisfy $\left|f_n^{\prime }\left(x\right)\right|\le g\left(x\right)$ for every $n\ge 1$ and a.e. $x\in ℝ\text{.}$
   Prove that there exists a subsequence ${\left(f_{n_j}\right)}_{j\ge 1}$ which converges uniformly on the entire real line.
5. Consider a sequence of functions $f_n\in L^1\left(ℝ\right)$ with ${\Vert f_n\Vert }_{L^1}\le C$ for every $n\ge 1\text{.}$ Define Prove that $f$ is Lebesgue measurable and ${\Vert f\Vert }_{L^1}\le C\text{.}$
6. Let $f:ℝ\to ℝ$ be an absolutely continuous function. Prove that $f$ maps sets of Lebesgue measure zero into sets of Lebesgue measure zero.
7. 1. If ${\left(f_n\right)}_{n\in {\mathbb{Z}}_{>0}}$ is a sequence of functions in $L^1\left([0,1]\right)$ such that ${\Vert f_n\Vert }_{L^1}\to 0$ prove that there exists a subsequence that converges pointwise for a.e. $x\in [0,1]\text{.}$
   2. Construct a sequence of measurable functions $f_n:[0,1]\to [0,1]$ such that ${\Vert f_n\Vert }_{L^1}\to 0$ but, for each $x\in [0,1]$ the sequence $f_n\left(x\right)$ has no limit.
8. For every (nonempty) open set $\Omega \subseteq {ℝ}^n$ and for $1\le p\le \infty ,$ prove that the space $L^p\left(\Omega \right)$ is infinite dimensional. Construct a sequence of functions ${\left(f_j\right)}_{j\in {\mathbb{Z}}_{>0}}$ such that $$\begin{array}{ccc}{\Vert f_j\Vert }_{L^p}& =& 1,\text{and}\\ {\Vert f_i-f_j\Vert }_{L^p}& \ge & 1\text{for all}\hspace{0.17em}i,j\in {\mathbb{Z}}_{>0}\hspace{0.17em}\text{with}\hspace{0.17em}i\ne j\text{.}\end{array}$$
9. Consider the set ${ℝ}^2$ with the partial ordering $x\le y$ if and only if $x_1\le y_1$ and $x_2\le y_2\text{.}$ Let $f:ℝ\to ℝ$ be a continuous, non decreasing function. Show that the set $$S=\text{Graph}\left(f\right)=\left\{(t,f\left(t\right))\hspace{0.17em}\right|\hspace{0.17em}t\in ℝ\}$$ is a maximal totally ordered subset of ${ℝ}^2\text{.}$ Is every maximal totally ordered subset obtained in this way?
10. Give a proof of the generalized Hölder inequality: If $1\le p_1,\dots ,p_m\le \infty$ and $\frac{1}{p_1}+\cdots +\frac{1}{p_m}=1$ and $f_k\in L^{p_k}\left(\Omega \right)$ for $k=1,2,\dots ,m$ then $${\int }_{\Omega }\left|f_1{f}_2\cdots f_m\right|dx\le \prod _{k=1}^m{\Vert f_k\Vert }_{L^{p_k}\left(\Omega \right)}\text{.}$$

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