# Math Help - Measurability

1. ## Measurability

Let $n,q \in \mathbb{N}$ such that $q \leq n$ and let $E_1,E_2,...,E_n$ be measurable subsets of $[0,1]$. Suppose that for each point $x \in [0,1]$, there are at least $q$ sets in $\{E_1,E_2,...,E_n\}$ that contain $x$. Prove that there exists $1 \leq i \leq n$ such that $m(E_i) \geq \frac{q}{n}$.

2. ## Re: Measurability

Originally Posted by Markeur
Let $n,q \in \mathbb{N}$ such that $q \leq n$ and let $E_1,E_2,...,E_n$ be measurable subsets of $[0,1]$. Suppose that for each point $x \in [0,1]$, there are at least $q$ sets in $\{E_1,E_2,...,E_n\}$ that contain $x$. Prove that there exists $1 \leq i \leq n$ such that $m(E_i) \geq \frac{q}{n}$.
Here's the basic idea. Suppose that $m(E_i)<\frac{q}{n}$ then $\sum_{i=1}^{n}m(E_i), but why is that stupid (hint: use the idea that rougly the $E_i$'s ' $q$-fold cover' $[0,1]$).

3. ## Re: Measurability

Okay solved it. Thanks!