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I have two questions to solve :

(1) If $E_1$ and $E_2$ are measurable, show that $|E_1 ∪ E_2| + |E_1 ∩ E_2| = |E_1| + |E_2|$

My Attempt :

We may assume that both $|E_1| , |E_2|\lt + \infty$, or else the result is trivially true.

Otherwise, since $$(E_1 \cup E_2) - E2 = E_1 - (E_1 \cap E_2),$$ $E_2 \subseteq (E_1 ∪ E_2)$, and $(E1 \cap E_2) \subseteq E_1$,

We have $|E1 ∪ E2| − |E2| = |E1| − |E1 ∩ E2|$ ,

We need that $|E_1| , |E_2| \lt + \infty $ in order to subtract measures. It is possible to do this problem without using any subtraction, though.

(2) Let $\{E_n\}_{n=1}^{\infty}$ be a sequence of measurable sets of $[0,1] \subset \mathbb{R}$ with $\sum_{n=1}^{\infty} \mu(E_n) \gt2$. Prove that there exists $E_i, E_j,E_k$ such that $\mu (E_i \cap E_j \cap E_k) \gt 0$

I don't know how to solve the second question, anyone know how to solve this question ? and how about question number 1, is it true ?

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  • $\begingroup$ I'd do the first by considering that $(E_1 \cap E_2) \sqcup (E_1 \setminus E_2) = E_1$ and $(E_1 \setminus E_2) \sqcup E_2 = E_1 \cup E_2$, where $\sqcup$ is the disjoint union. $\endgroup$ – B. Mehta Nov 27 '17 at 4:57
  • $\begingroup$ Is $\mu$ the Lebesgue’ measure ? $\endgroup$ – Youem Apr 11 '18 at 5:05

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