## Bressan problems

Last update: 18 March 2014

## Bressan problems

1. Let $\left\{{A}_{i} | i\in I\right\}$ 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 $B\left(x,\rho \right)\subseteq {A}_{j}\text{.}$
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 ${\left({x}_{{n}_{j}}\right)}_{j\in {ℤ}_{>0}}$ one can extract a further subsequence converging to $x\text{.}$
2. If $d\left({x}_{m},{x}_{n}\right)\ge \delta$ 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 ${\left({x}_{{n}_{j}}\right)}_{j\in {ℤ}_{>0}}$ such that $limsupj,k→∞ d (xnj,xnk) <ε.$ Then the sequence admits a convergent subsequence.
3. Consider the function $f(x)= { 1x(ln x)2, if 0 Let ${f}_{\epsilon }\doteq {J}_{\epsilon }*f$ be the corresponding mollifications, and let $F(x)≐sup0<ε<1 fε(x).$ 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 ${‖{f}_{\epsilon }-f‖}_{{L}^{1}\left(ℝ\right)}\to 0\text{.}$
4. Let ${f}_{n}:ℝ\to ℝ$ for $n\in {ℤ}_{\ge 1},$ 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 $|{f}_{n}^{\prime }\left(x\right)|\le g\left(x\right)$ 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 ${‖{f}_{n}‖}_{{L}^{1}}\le C$ for every $n\ge 1\text{.}$ Define $f(x)≐ { limn→∞fn(x), if the limit exists, 0, otherwise.$ Prove that $f$ is Lebesgue measurable and ${‖f‖}_{{L}^{1}}\le C\text{.}$
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.
1. If ${\left({f}_{n}\right)}_{n\in {ℤ}_{>0}}$ is a sequence of functions in ${L}^{1}\left(\left[0,1\right]\right)$ such that ${‖{f}_{n}‖}_{{L}^{1}}\to 0$ prove that there exists a subsequence that converges pointwise for a.e. $x\in \left[0,1\right]\text{.}$
2. Construct a sequence of measurable functions ${f}_{n}:\left[0,1\right]\to \left[0,1\right]$ such that ${‖{f}_{n}‖}_{{L}^{1}}\to 0$ but, for each $x\in \left[0,1\right]$ the sequence ${f}_{n}\left(x\right)$ has no limit.
7. 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 ${\left({f}_{j}\right)}_{j\in {ℤ}_{>0}}$ such that $‖fj‖Lp = 1,and ‖fi-fj‖Lp ≥ 1for all i,j∈ℤ>0 with i≠j.$
8. 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 $S=Graph(f)= {(t,f(t)) | t∈ℝ}$ is a maximal totally ordered subset of ${ℝ}^{2}\text{.}$ Is every maximal totally ordered subset obtained in this way?
9. 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 $∫Ω|f1f2⋯fm| dx≤∏k=1m ‖fk‖Lpk(Ω).$