Outer measure > $0$?

Question

Let's say we have $A \subset I_0$ as an arbitrary set such that

$Int(A) \neq \emptyset$

My question is: is $\mu^* (A)$ always non-negative/positive?

Answer 1

It depends on what kind of measure $\mu^*$ is. I assume $\mu^*$ is the outer Lebesgue measure and $I_0 \subset \mathbb R$ an arbitrary interval. Since $\mathring A$ is open and $\mathring A \neq \emptyset$, you find a non-empty open interval $(a,b) \subset \mathring A$. Now we have $$ 0 < \underbrace{\mu^*((a,b))}_{=b-a} \leq \mu^*(\mathring A) \leq \mu^*(A) \; .$$