Note on the Outer Measures of Images of Sets

Abstract

Let $f$ be a real function on ${\mathbb R}$, let $\{I_v\}$ be a family of intervals covering a set $E$ such that $m(E \cap I_v) \ge m\bigl (f(E \cap I_v)\bigr )$ for each $I_v$. We prove that $m\bigl (f(E)\bigr ) \le 2 \cdot m(E)$. No coefficient smaller than $2$ will suffice here in general.