If $A$ is bounded then the outer measure of $A$ is finite

Question

I got entangled with the following question:

I need to prove or disprove:

Let $A \subseteq \mathbb R$

1. If $A$ is bounded then $m^*(A) \lt {\infty }$
2. If $m^*(A) \lt {\infty }$ then $A$ is bounded

Answer 1

The first statement is true (I will leave the writing of the proof to you), assuming $m^*$ means the Lebesgue outer measure on $\mathbb{R}$. Note that:

• $A$ is bounded (i.e. we have $A\subseteq [-M,M]$ for some $M>0$).

• For closed intervals $I=[a,b]$ (which are measurable) we have $m^*(I)=\ell(I)=b-a$.

• The outer measure is monotone.

Conversely, the second statement is not true. For instance, let $A=\mathbb{Q}$ (again, fill in the gaps).