Is outer measure always nonnegative?

Question

I know that if you take the measure of the null set, the measure is 0.

But say you take a set where the interior of the set is not the empty set. Then is the outer measure of the set positive, and is there ever a case where the measure is negative? Furthermore, why does it follow that the outer measure is 0 if the set is countable? That seems counter-intuitive to me.

Answer 1

Hint: For the countable set $\{x_n\}$: surround the point $x_n$ with an interval of length $\epsilon/2^n$. The sum of the lengths of the intervals is $$\epsilon + \epsilon/2 + \epsilon/2^2 + \cdots = 2 \epsilon$$