The conversation between human and AI assistant.
How do I solve the following problem in measure theory ?
Here it is: Show that if $\{E_j\}$ is a countable collection of measurable sets in $\mathbb{R}^d$, then $\bigcap\limits_{j=1}^{\infty} E_j$ is measurable.
Okay. First, we know that every measurable set $E \subset \mathbb{R}^d$ can be written as a countable union of closed sets: $E = \bigcup\limits_{n=1}^{\infty} F_n$, where each $F_n$ is closed.