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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.2012-05-06

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