Saxe's "Beginning Functional Analysis" is the textbook for a course which covers measure theory I am currently sitting. On page 38, in going on to introduce the Lebesgue measurable sets of $\mathbb R^n$, the following is given:
"We let $\mathcal M_\mathcal F$ denote the collection of subsets $A\in\mathbb R^n$ such that $D(A_k,A)\to0$ as $k\to\infty$ for some sequence of sets $A_k\in\mathcal E$. We let $\mathcal M$ denote the collection of subsets of $\mathbb R^n$ that can be written as a countable union of sets in $\mathcal M_\mathcal F$."
Here $D(A,B)=m^*(A\Delta B)$ is the "distance from $A$ to $B$", for $A,B\in\mathbb R^n$, where $A\Delta B$ denotes the symmetric difference of $A$ and $B$, and $m^*$ is the outer measure of $A\Delta B$. The collection $\mathcal E$ is that of all finite unions of disjoint intervals of $\mathbb R^n$.
My problem: I'm finding it hard to understand what, exactly, $\mathcal M_\mathcal F$ is, and, on an intuitive level, what it means for a set $A$ to be in $\mathcal M_\mathcal F$. Furthermore, what is special about being in $\mathcal M_\mathcal F$? What characterises those sets which make up $\mathcal M_\mathcal F$ differently from an arbitrary set of $\mathbb R^n$? Could somebody help to give me an intuitive overview here?
Imagine that $\mathcal{E}$ is the collection of all open boxes of the form $(a_{1},b_{1})\times\cdots\times(a_{n},b_{n})$, $\mathcal{M}_{\mathcal{F}}$ is the collection of all open sets or closed sets, one can imagine that $\mathcal{M}_{\mathcal{F}}$ consists of the Borel sets. Finally $\mathcal{M}$ may resemble of Lebesgue measurable sets.