\[ \newcommand{\C}{\mathbb{C}} \newcommand{\haar}{\mathsf{m}} \newcommand{\B}{\mathscr{B}} \newcommand{\D}{\mathscr{D}} \newcommand{\P}{\mathcal{P}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\g}{>} \newcommand{\l}{<} \newcommand{\intd}{\,\mathsf{d}} \newcommand{\Re}{\mathsf{Re}} \newcommand{\area}{\mathop{\mathsf{Area}}} \newcommand{\met}{\mathop{\mathsf{d}}} \newcommand{\emptyset}{\varnothing} \DeclareMathOperator{\borel}{\mathsf{Bor}} \DeclareMathOperator{\baire}{\mathsf{Baire}} \newcommand{\symdiff}{\mathop\triangle} \]

1. Fix $E \subset \R$. Define what it means for a function $f : E \to \R$ to be measurable. For your definition, prove or disprove the following statement.
   
   Statement Whenever $f : \R \to \R$ is Borel measurable the restriction of $f$ to $E$ is measurable.
2. Fix a mesaure $\mu$ on $(\R,\borel(\R))$. Prove that the complement of \[ \{ x \in \R : \mu(x-r,x+r) > 0 \textsf{ for every } r > 0 \} \] is assigned zero measure by $\mu$.