Outer Measure Definiton

Question

$\textbf{Definition:}$ Let $A\in \mathcal{P}(\mathbb{R})$. In the real analysis book I am using by Royden, he says consider $\lbrace I_n \rbrace$ to be a countable collection of open intervals which cover $A$ (aka $A\subseteq \bigcup I_n$) and denotes the outer measure to be as follows: $m^*(A)=\underset{A\subseteq\cup I_n}{\inf}\sum l(I_n)$.

Both the notation he uses and the wording (particularly with the word $\textbf{consider}$) is confusing me. Does he mean let $\lbrace I_n \rbrace$ cover $A$ by an arbitrary countable set of open intervals? Does he mean $\lbrace I_n \rbrace$ is associated with set builder notation which is left out? I tried to rewrite his definition below as best I could below and am wanting to see if the definition I am using is correct.

$\textbf{Question:}$ Will the definition I made up below match up with his definition?

$\textbf{Made Up Definiton:}$ Let $A\in \mathcal{P}(\mathbb{R})$. Then, we denote the outer measure of $A$ as follows: $m^*(A):= \inf\{{\sum_{j\in J}{l(I_j)|\exists \text{ a countable collection of open intervals }\lbrace I_j \rbrace_{j\in J}: A \subseteq \bigcup_{j\in J} I_j}\}}$.

Answer 1

They're equivalent.

The pedantic way of expressing this is the following:

Define $m^*:\wp(\mathbb R)\to[0,\infty]$ by

$$A\mapsto\inf\left\{\sum_{j\in J}l\left(I_j\right):\left\{I_j\right\}_{j\in J}\text{ is a countable collection of open intervals that covers }A.\right\}.$$

Answer 2

Observe that there always exist a countable collection of open intervals covering $A$. The existential qualifier in your definition does not make much sense to me. What the definition does is consider all countable collections of open intervals $\lbrace {I_j}\rbrace$ covering $A$ and take the infimum of the sum of the lengths of the intervals in the collection. This may be what you are looking for:

$$ m^*(A)=\inf\Bigl\lbrace\sum_{j\in J}l(I_j):\lbrace I_j\rbrace\text{ is a countable collection of open intervals covering }A\Bigr\rbrace. $$