Lebesgue measurable subsets of $[0,1]$

Question

Problem

Let $E_1,...,E_n$ be measurable sets contained in the interval $[0,1]$. If for each $x \in [0,1]$, the set $A_x=\{k: 1 \leq k \leq n \space \text{and} \space x \in E_k\}$ has at least $q$ elements , show that there is $k \in \{1,...,n\}$ with $|E_k| \geq \dfrac{q}{n}$

I could only deduce the following things:

1) $[0,1]=\bigcup_{i=1}^n E_i$

2) There is some $j$ such that $|E_j| \geq \dfrac{1}{n}$ (this can be deduced from 1.)

If I divide the interval $[0,1]=[0,\dfrac{1}{n}) \cup ... \cup [\dfrac{n-1}{n},1]$, maybe I could show that there are $q$ intervals that belong to one of the $E_k$'s

I would appreciate hints to solve the problem rather than a complete answer. Thanks in advance for your help.

Answer 1

Let $1_{E_k}$ be the characteristic function of $E_k$ and let $f=\sum_{k=1}^n 1_{E_k}$. What can you say about the integral of $f$?

Answer 2

We're going to prove that $\sum_{k=1}^n |E_k|\geq q$, from which the result will follow.

Consider the indicator functions ${\bf 1}_{E_k}$, and let $f=\sum_{k=1}^n {\bf 1}_{E_k}$. Our hypothesis says that $f(x)\geq q$ for each $x$. Integrating both sides from $0$ to $1$ w.r.t. Lebesgue measure yields $\sum_{k=1}^n |E_k|\geq q$.