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Let $k, n \in \mathbb{N}: k ≤ n$ and let $E_1, \dots , E_n \subseteq [0, 1]: ((\forall i, E_i$ is Lebesgue measurable subset$)$ & $( x \in [0, 1] \Rightarrow |\{E_i: x \in E_i, i \leq n \}| \geq k))$; (that is: $x$ belongs to at least $k$ of the sets $E_1,\dots,E_n $).

Show that there exists $1 ≤ i_0 ≤ n: λ(E_{i_0}) \geq k/n$.

I am not sure where to start.

user332420
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1 Answers1

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Hint:

$\sum_{i=1}^{n}1_{E_{i}}\geq k1_{[0,1]}$ so taking the integral on both sides we find by linearity of integrals: $$\sum_{i=1}^{n} \lambda (E_{i}) \geq k$$

ForgotALot
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drhab
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  • So, if I assume that, $\forall i \in {1,\dots, n }, λ(E_i) < k/n$, then their sum will be less than $k$, which is a contradiction. Thus, at least one $E_i$ (denoted $E_{i_0}$) has measure that is not less than $k/n$. Nice! Thanks! – user332420 Apr 18 '16 at 23:19