Approximating measurable sets of infinite measure by open sets

Question

Following a question posted here: Approximating measures by open sets and compact sets.

I wanted to ask, if I am given a measurable set $E\subseteq \mathbb{R}$ s.t. $m(E)=\infty$, then how can I find an open set $O\supseteq E$ s.t. $m(O- E)<\epsilon$ for a given $\epsilon>0$?

Answer 1

Given the formulation of your question, I assume that you already know the claim for sets of finite measure.

Now, for $n\in \Bbb{N}$, there is an open set $O_n \subset (-n,n)$ with $O_n \supset E_n := E\cap(-n,n)$ and $\mu(O_n \setminus E) = \mu(O_n \setminus E_n)<\epsilon/2^n$. Here,I used $O_n \subset (-n,n)$ to get the first equality.

Now, $O:=\bigcup O_n$ is open with $O\supset E$ (why?) and $$ \mu (O\setminus E)\leq \sum_n \mu(O_n \setminus E)\leq \sum_n \epsilon/2^n \leq \epsilon. $$