Let $A_1, A_2, \dots$, be measurable sets, not necessarily disjoint, such that each set is a subset of $\mathbb{R}^n$. If $m(A_i \cap A_j) = 2$ for all $i, j \in \mathbb{N}$, then how can we prove that a finite intersection of the first $n$ sets has measure 2? What about an infinite intersection of all the sets? Here $m$ denotes Lebesgue measure.
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Does it apply if $i=j$? $\mu(A_i \cap A_i) = 2$? – Michael Mar 15 '20 at 17:26
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Yes, I believe it does. – Nathan Park Mar 15 '20 at 19:10
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1Then, for example, can you prove $\mu(A_i \cap A_j^c) = 0$ for $i \neq j$? – Michael Mar 15 '20 at 19:18
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https://math.stackexchange.com/questions/234292/continuity-from-below-and-above This may help – Math1000 Mar 15 '20 at 21:43
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Let $m$ be a positive integer that is greater than or equal to 2.
Prove that $\mu(A_1)=2$.
Prove that for all $j \in \{2, 3, 4, …\}$ we have $\mu(A_1 \cap A_j^c) = 0$.
Prove that $\mu\left(A_1 \cap (\cap_{j=2}^m A_j)\right) = 2$.
Repeat the same technique to compute $\mu\left(A_1 \cap (\cap_{j=2}^{\infty} A_j)\right)$.
Can you do one or more of these steps?
Michael
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I can do them all using demorgan's laws and properties of measure, except the last one. When there are infinitely many sets, how is it different than when there are finitely many sets? In other words what should I be careful about? – Nathan Park Mar 15 '20 at 22:29
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