Why $\{x: \limsup_nf_n(x)>a\}=\bigcap_{n=1}^{\infty}\bigcup_{k \geq n}^{\infty}\{x:f_n(x)>a\}$ is measurable?

Question

Why $$\{x: \limsup_nf_n(x)>a\}=\bigcap_{n=1}^{\infty}\bigcup_{k \geq n}^{\infty}\{x:f_n(x)>a\}$$ $$\{x: \liminf_nf_n(x)>a\}=\bigcup_{n=1}^{\infty}\bigcap_{k \geq n}^{\infty}\{x:f_n(x)>a\}$$ are measurable?

If I know $\{x:f_n(x)>a\}$ is measurable, then the infinite union is measurable. But why $\cap_{n=1}^{\infty}$ is still measurable?

Answer 1

Intersection of any sequence of measurable sets is measurable since $\bigcap_n E_n=(\bigcup_n E_n^{c})^{c}$. (Complements of measurable sets and countable unions of measurable sets are measurable).