$A \subseteq \mathbb R^n$ Lebesgue measurable of positive measure . Then for every $m \in (0,L(A)] , \exists A_m \subseteq A $ such that $L(A_m)=m$?

Question

Let $A \subseteq \mathbb R^n$ be Lebesgue measurable of positive Lebesgue measure . Then how to show that for

every $m \in (0,L(A)] , \exists A_m \subseteq A $ , $A_m$ Lebesgue measurable , such that $L(A_m)=m$ ? ( Here $L$ denotes the Lebesgue measure)

Answer 1

Hint. Study the function $F(r) = \mu(A \cap B_r(0))$ defined for $r \in (0,+\infty)$.