I was wondering if anybody can help me figure out what the following exercise is supposed to say, because as it's stated I'm nearly certain this is wrong.
Let $\mathcal{E}$ be the set of all open intervals on $\mathbb{R}$. Define $\rho((a,b))=(b-a)^2$. Let $\mu^*$ be the outer measure induced by $\rho$. Prove that $\mu^*=0$.
When the exercise refers to the outer measure induced by $\rho$, I believe it is referring to proposition 1.10 from Folland (which I'm pretty sure also has a typo).The proposition (with the typo) is stated as:Let $\mathcal{E}\subset \mathcal{P}(X)$, and $\rho: \mathcal{E}\to[0,\infty]$ be such that $\emptyset \in \mathcal{E}$ and $X\in\mathcal{E}$. For any $A\subset X$, define$$\mu^*(A)=\inf\Bigg\{\sum_{j=1}^\infty\mu\left(E_j\right): E_j\in\mathcal{E} \text{ and } A\subset\bigcup_{1}^\infty E_j\Bigg\}.$$Then $\mu^*$ is an outer measure.
I think the typo in the statement of prop 1.10 is in the scope over which the infimum is taken. I believe $\sum_{j=1}^\infty\mu\left(E_j\right)$ is supposed to say $\sum_{j=1}^\infty\rho\left(E_j\right)$, but I don't see that in any of the errata I could find.
Regarding the exercise, the statement seems false to me, since I can just take $A=(0,1)$, then $\mu^*((0,1)) = (1-0)^2=1.$
Can anyone tell what the correct result is supposed to be? Or if I'm the one who's wrong and the result is actually correct?
Thanks so much.
