I am trying to show the two following properties of measurable sets:
(i) If $E \subset \mathbb R^n$ is measurable, then for each $0\leq r \leq |E|$, there is $F \subset E$ measurable such that $|F|=r$.
(ii) Let $E \subset \mathbb R^n$ be a measurable set, of finite measure. Show that for each $n \in \mathbb N$, there are $n$ disjoint subsets of $E$,$(E_j)_{1\leq j \leq n}$, such that $|E_j|=\dfrac{|E|}{n}$ for all $j$.
I am pretty lost in both items, I've seen some properties of measurable sets but I don't know which could be useful to show each of these properties. I would appreciate suggestions or hints.
I will attempt to give you hints without giving away the entire solution. Please let me know if I give too much / too little advice.
(i) Let $B_r$ denote the open ball of radius $r$ centered at the origin. Consider the function $f(r) = |B_r \cap E|$ (I suppose that $| \cdot |$ denotes the Lebesgue measure of a set). Show that $f$ is continuous.
(ii) Use the continuity of the same function $f$ above to derive this conclusion as well.
Hopefully this will give a nudge in the right direction!