Week 2 Coursework Test

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.
Fix a mesaure $μ$ on $(R, Bor (R))$ . Prove that the complement of ${x \in R : μ (x - r, x + r) > 0 for every r > 0}$ is assigned zero measure by $μ$ .