Computing an outer measure

Question

How do you actually compute an outer measure? I know the definition. It is: $$\mu^*(B):=\inf\left(\sum\limits_{k=1}^n \mu(I_k):B \subset \bigcup\limits_{k=1}^n I_k\right)$$ But how do you use this to actually compute something? For example: $$\mu^*\left(\left(1+\frac{1}{n}\right)^n \mid n \in \mathbb{N}\right)$$ Is it literally just the infimum of that function of n? The premeasure is defined as $B=[a,b]$ and $A \subset B$.

Answer 1

$\mu^*$ is the outer measure of a set, and it is defined as the infimum of the sums of the premeasures over all possible coverings of your set.

So, imagine your set is a stain on a wall. Your premeasure is a paintbrush.

The outer measure is defined as the smallest possible way you can arrange brushstrokes to cover the stain. How to compute it depends greatly on your domain and your premeasure. It's usually most useful to think of the outer measure as a function that has certain properties (which we might want to use to extend to a complete measure).

So to answer your specific question, if I am interpreting it right, we cannot compute $\mu^*\left(\left\{ \left(1+\frac1n\right)^n : n \in \mathbb{N}\right\}\right)$ in this sense until you define a premeasure of some sort.