In Measure Theory, the regularity of measures is defined as:
In any topological space $ \left(\Omega,\tau\right) $ , $ \mu $ is a measure on it; $ \mu $ is inner regular:
$$ \forall A\subseteq\Omega, \mu\left( A \right) = \sup\left\{ \mu\left(K\right) ; K\subseteq A , K \text{ is compact} \right\} $$
$ \mu $ is outer regular:
$$ \forall A\subseteq\Omega, \mu\left( A \right) = \inf\left\{ \mu\left(O\right) ; A\subseteq O , O \text{ is open} \right\} $$
and $ \mu $ is regular if and only if it is inner and outer regular simultaneously.
These definitions helped connect the measure structure and topological structure in a space $ \Omega $ that has these two structures.
But why? Why do we have these concepts rather than not?
I, as a noob in measure theory, the only reason I know is that the Lebesgue measure in $ \mathbb{R}^n $ fulfils these definitions, thus we can approximate any subset by open or compact (in $ \mathbb{R}^n $ those who will be equivalent with bounded and closed) sets in the meanings of measure, and I do know that compactness is so good since there exists naturally a finite cover and covers are important in measure theory, but I'm not entirely convinced buy just that. So my questions are:
Is it the main reason for defining the regularity of measures? If so, then why is it so important to approximate sets by open or closed sets?
If not, what's the main motivation for defining regular measures?
Why do we call such measures regular?
The last question is a little bit more terminology or etymology, not strictly in the domain of Maths, I don't know if it's appropriate in this forum, but I insist on asking you for help. Thank you very much in advance for any comment on any of my questions, if my questions aren't so stupid or untimely...
Nice day~