A subset E of X is said to be $\mu^*$ measurable if
$\mu^*(A) = \mu^* (A \cap E) + \mu^*(A \backslash E)$ for all subsets A of X.
In other words what this says is a set E is $\mu^*$ measurable if E and its compliment are sufficiently separated that they divide an arbitrary set A additively.
What is the intuition behind this definition? What is this saying?
I know $\mu^*$ means outer measure but what does the term $\mu^*$ measurable mean?
Does it just mean that the outer measure of a subset can be found?
Best Answer
People would just say a set is measurable if Caratheodory condition is satisfied. I feel you always confuse terms. I guess you need change a book, like Royden's
As regard the intuition of Caratheodory condition, it bothered me (and still) and I think it can be a quite deep problem:-( Here is the best explanation I can find here