suppose $\{E_n\}_n$ is a sequence of measurable sets, $J$ is a positive integer, $G$ is the set which contains the points that at least belong to $J$s $E_n$, i.e.
$$G= \{ x : \exists n_1 < \dots < n_J | x \in E_{n_1} \cap \dots \cap E_{n_J}\}$$ Prove that $G$ is measurable and $m(G)\le \frac{\sum m(E_n)}{J}$
I had proof the first part by defining $f_n(x)=X_{E_n}(x)$(characteristic function of $E_n$),and $G= \{x: \sum f_n(x) \ge J \}$ but the second part is hard to prove.
I assume that: $$G:=\{x\in X\mid |\{n\mid x\in E_n\}|\geq J\}$$ Observe that: $$1_G(x)=1\iff x\in G\iff\sum 1_{E_n}(x)\geq J$$This can also be expressed by$$J1_G(x)\leq\sum 1_{E_n}(x)$$
On both sides take integrals to arrive at:$$Jm(G)\leq\sum\mu(E_n)$$