Show that the outer measure for $[a,b)\subset \mathbb{R}$ is $|[a,b)|=b-a$.

Question

Show that the outer measure for $[a,b)\subset \mathbb{R}$ is $|[a,b)|=b-a$.

I can show that $|(a,b)|=|[a,b]|=b-a$, but I am not sure where to start with a half-open interval. I can assume there's an open cover $\mathcal{I}$ for $[a,b)$, but what next? Any help is appreciated.

Answer 1

Use the fact that $$(a,b)\subseteq [a,b) \subseteq [a,b],$$ in conjunction with what you showed about $|(a,b)|=|[a,b]|$