\(\newcommand\notesclassalt[2]{\if\smallformat1#2\else#1\fi} \newcommand\lectureend[1]{% \noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}} \marginpar{\raggedleft#1}% } \newcommand\isdefinition[1]{\textbf{#1}} \newcommand\f\frac \newcommand\intd{\mathop{}\!\mathrm{d}} \newcommand\ind[1]{1_{#1}} \newcommand\seq\coloneqq \newcommand\qes{=:} \newcommand\loc{\mathrm{loc}} \newcommand\A{\mathcal A} \newcommand\R{\mathcal R} \newcommand\Q{\mathcal Q} \newcommand\B{\mathcal B} \renewcommand\P{\mathcal P} \newcommand\spt[1]{\mathrm{spt}(#1)} \newcommand\var{\mathop{\mathrm{var}}} \newcommand\diam{\mathop{\mathrm{diam}}} \newcommand\dist{\mathop{\mathrm{dist}}} \newcommand\dmo{\mathrm d} \newcommand\dm[2]{\dmo(#1,#2)} \newcommand\tb[1]{\mathop{\partial{}}{#1}} \newcommand\mb[1]{\mathop{\partial_*}{#1}} \newcommand\mbe[1]{\mb{(#1)}} \newcommand\mbb[1]{\mb{\Bigl(#1\Bigr)}} \newcommand\tbe[1]{\tb{(#1)}} \newcommand\tbb[1]{\tb{\Bigl(#1\Bigr)}} \newcommand\rb[1]{\mathop{\partial^*}{#1}} \newcommand\tc[1]{\overline{#1}} \newcommand\mc[1]{\tc{#1}^*} \newcommand\ti[1]{\mathring{#1}} \newcommand\tia[1]{#1^{\mathrm o}} \newcommand\tim[1]{\tia{(#1)}} \newcommand\tib[1]{\tia{\bigl(#1\bigr)}} \newcommand\mi[1]{\ti{#1}^*} \newcommand\mia[1]{#1^{\mathrm o*}} \newcommand\mim[1]{\mia{(#1)}} \newcommand\mib[1]{\mia{\bigl(#1\bigr)}} \newcommand\smg{\mathcal{H}} \newcommand\smo{\smg^{d-1}} \newcommand\sm[1]{\smg^{d-1}(#1)} \newcommand\smb[1]{\smg^{d-1}\Bigl(#1\Bigr)} \newcommand\smi[1]{\smg^0(#1)} \newcommand\smib[1]{\smg^0\Bigl(#1\Bigr)} \newcommand\lmr[1]{|#1|} \newcommand\lmg{\mathcal{L}} \newcommand\lmog{\lmg_*} \newcommand\lm[1]{\lmg(#1)} \newcommand\lmb[1]{\lmg\Bigl(#1\Bigr)} \newcommand\lmib[1]{\lmio\Bigl(#1\Bigr)} \newcommand\lmo[1]{\lmog(#1)} \newcommand\lmob[1]{\lmog\Bigl(#1\Bigr)} \newcommand{\mres}{\!\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.1ex}\!} \newcommand\restrictzero[2]{#1\,\mres#2} \newcommand\D{\mathbb D} \newcommand\Dq{\mathcal D} \newcommand\M{\mathcal M} \)
Is Lebesgue measure a Radon measure? Is Lebesgue outer measure a Radon outer measure?
Do the annuli \(A_n=\tc {B(x,n)}\setminus B(x,n-1)\) form a decomposition, i.e. is it true that \(Ω=A_1∪A_2∪…\) and for any \(n\neq k\) we have \(A_n∩A_k=∅\)?
Is it true, that for any \(E⊂Ω\ni x\) and any Radon measure \(μ\) that \(μ(E\setminus B(x,n))→0\) as \(n→∞\)?
Let \(μ\) be the counting measure on \(ℝ\), i.e. \(μ(E)\) equals the number of elements in \(E\) if \(E\) is finite, and \(μ(E)=∞\) otherwise. Is \(μ\) a Radon measure? Is \(μ\) a Radon outer measure?
Does ?? hold for the counting measure?
Is it true that for any measurable \(A⊂ℝ^d\) with \(\lm A<∞\) and \(ε>0\) exists a compact set \(K⊂A\) with \(\lm {A\setminus K}<ε\)? What, if \(\lm A=∞\)?
Let \(E⊂Ω\) be measurable. Is the characteristic function given by \[ \ind E(x) = \begin {cases} 1&x∈E \\ 0&x\notin E \end {cases} \] measurable?
Are polynomials Lebesgue measurable functions?