Define $A_k=\{x\in A: |x|\leq k\}$, then $\{A_k:k\in\mathbb{N}\}$ is measurable also.

Question

Let $A\subset\mathbb{R}^n$. Assume $A$ is Lebesgue measurable, and unbounded. Define $A_k=\{x\in A: |x|\leq k\}$, then $\{A_k:k\in\mathbb{N}\}$ is an increasing sequence of bounded sets whose union is $A$. But why are they measurable also?

Answer 1

They are closed balls intersected with A, $A_{k} =\overline{B(0,k))}\cap A$, which are measurable.. Intersection of measurable sets is measurable!