[Math] Measurable sets are approximately open set.

Question

I am new in measure theory and studying N.L.Carothers Real Analysis book. It is said in this book that measurable sets are approximately open. How is it this? But it is also said that we say that a set $E$ is measurable if, for each $\epsilon>0$ we can find a closed set $F$ and an open set $G$ with $F\subset E\subset G$ such that $m^{*}(G\setminus F)<\epsilon.$ According to this definition can we say that measurable sets are approximately closed? I am confused please suggest me. I am learning measure theory by self-study. Please help me. Thanks a lot.

Answer 1

A measure which satisfies the fact that given $E\in \mathcal S$ (where $\mathcal S$ is the measure space), and given any $\epsilon$, we can find an open set $O$ and closed set $C$ with $C\subset E\subset O$ and $\mu(O-C)<\epsilon$ is called a "regular" measure.

I think Wikipedia may give some insight towards your answer with its slightly different definition. What it is basically saying is that an "inner regular" measure can be approximated by closed sets, and an "outer regular" measure can be approximated by open sets. So "regular" measure can basically be approximated by both open and closed sets. The Lebesgue measure is a regular measure.

Note: Be careful by what you mean by $m^*$, I usually see this as denoting the outer measure, but it can mean many other things. The outer measure (which is used in constructing the Lebesgue measure--see Caratheodory's theorem), is itself not necessarily a well defined measure (depending on the space on which you define it.