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.
Finite and infinite intersection of measurable sets
lebesgue-measuremeasure-theory
Best Answer
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?