Show that, for all $\delta < \mu(S) $, $\delta >0 $, exists a subset $T$ of $S$ such that $\mu(T) = \delta$.

Question

Let $ \mu $ the lebesgue measure on $\mathbb{R}$ and let $S$ a subset of $\mathbb{R}$ with $\mu(S) > 0 $.

$(a)$ Show that, for all $\delta < \mu(S) $, $\delta >0 $, exists a subset $T$ of $S$ such that $\mu(T) = \delta. $

(We can start with S bounded)

$(b)$ For all Borel measure (a) is true?

I started from the definitions of measure (outer) Lebesgue and use union of intervals but I don't know how to proceed

Answer 1

For (b), consider the Dirac measure (point mass at the origin) as a counterexample.

For (a), consider the map

$$ F :(0,\infty), t \mapsto \mu((-t,t)\cap S). $$

Show that $F$ is increasing and continuous with $F(t)\to \mu(S)$ as $t\to\infty$.

Why does that help you?