Lately I started learning measure theory. I have a question about the definition of when a set is outer measurable. The definition states:
Definition: Let $\mu^{\ast} $ be an outer measure. A set $A \subset X$ is $\mu^{\ast}$ measurable if $\mu^{\ast}(E)=\mu^{\ast}(E \cap A)+\mu^{\ast}(E \cap A^{c})$ $\forall E \subseteq X$.
After some searching, I found this answer in this post Intuition behind the Caratheodory’s Criterion of a measurable set
Which states that if we assume $\mu^*(X)< \infty$
Then we can define an inner measure $\mu_{\ast}$ on $X$ by
(1) $\mu_{\ast}(E)=\mu^{\ast}(X)-\mu^{\ast}(E^c)$
In the post it is further statet that, by being able to define an inner measure through the outer measure makes the inner measure redundand.
My first question is, going with (1), how does one end up with $\mu^{\ast}(E)=\mu^{\ast}(E \cap A)+\mu^{\ast}(E \cap A^{c})$? My second question:
When I think about the problem to measure something, (especially remembering the Riemann integral) the way to do would be by defining an outer measure $\mu^{\ast}$ and an inner measure $\mu_{\ast}$ by
using the infimum and the supremum. And if $\mu_{\ast}(A)=\mu^{\ast}(A)$, then the set $A$ would be called measurable.
But in measure theory, it seems that we define an outer measure on a collection of sets. Then we consider the collection of all $\mu^{\ast}$ measurable sets $\mathcal{A}$ and restrict $\mu^{\ast}$ to $\mathcal{A}$. While I do (hope) to understand the procedure mathematically, I would like to get some (probably geometric) intuition on this. I have the feeling that there is a lot of redundant stuff that measure theory leaves out.
