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Assume $A_j,j\geq 1,j\in\Bbb N$ are measurable sets. Let $m \in N$, and let $E_m$ be the set defined as follows : $x \in E_m \Longleftrightarrow x$ is a member of at least $m$ of the sets $A_k$.

I wanna know how to prove that

  1. $E_m$ is measurable.
  2. $m\lambda(E_m)\le\sum^{\infty}_{k=1}\lambda(A_k)$.

It's hard to me. Help me T.T

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For the first question, try to write $E$ as a countable union involving the $A_j$, using the fact that the subsets of $\Bbb N$ which have $m$ elements is countable. – Davide Giraudo May 6 '12 at 12:54

Call $S=\sum\limits_{n=1}^{+\infty}\mathbf 1_{A_n}$. Here are some hints:

  • Show that $S$ is a measurable function (possibly as the pointwise limit of a sequence of measurable functions). Note that $E_m=[S\geqslant m]$, and deduce item 1.
  • Compute the integral $I$ of $S$ in terms of the sequence $(\lambda(A_n))_n$. Show that Markov's inequality reads $m\lambda(E_m)\leqslant I$ in the present situation, and deduce item 2.
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