Please help with definition for outer measure? The following definition is taken from page 29/Folland 's Real analysis.

In my understanding, this definition said :  the outer measure of any set A belongs to X is the largest lower bound of the set (Let say this set is S) which contains the value of the summation of function rho (actually this is rho not mu, that was a printing mistake, rho is the set function whose domain is the subset E of the power set of X) for sets Ej , these sets Ej are covers for set A. 
What I don't understand here is about that set S which contains those summations. Would you please show me the members which is contained in S ? Is that the { sum rho(E1 to En} , sum rho(E1 to En+1), sum rho (E1 to En+2) ,... and so on)  } . Then after I figure all the members in this set S, then I can take the largest lower bound of these members to return the outer measure right ? . Sorry for the long and silly questions. But I am new to proof based math. 
 A: You are not reading the definition quite right.  The outer measure is the smallest, upper bound of a set, where we compute the upper bound by taking a countable collection of sets from $\mathcal{E}$ and adding their measures.
The set $S$ that you point to is a collection of real numbers, possibly including $\infty$.
The idea here is that you might have lots of sets whose measure you know—like the open interval $(a,b)$ for example, which we guess should have length $b-a$—and you are trying to come up with a good definition of measure for any set.  It is not quite possible to do this, but you can define an outer measure, which is the best possible upper bound for the measure of your set $A$.
So $S$ is a collection of every real number that we could possibly get when covering $A$ by sets in $\mathcal{E}$.  For example, say we want to compute $\mu^* (\mathbb{R})$ using this definition, with the measure on open intervals I mentioned earlier.  Then, for example, $(-1,1), (-2,2), (-3,3), \ldots$ is a cover of $\mathbb{R}$, and the sum of its measures is $2+4+6+\ldots = \infty$.  So in this case, $\infty$ is an element of $S$.
The "inf" on the outside means that we should take the infimum, or the least upper bound, of the real numbers in $S$.  If you play around with the example in the last paragraph, I think you'll find that $\infty$ is the only element in that set, so we come up with the intuitive result $\mu^* (\mathbb{R}) = \infty$.
Similarly, we can calculate $\mu^* ([0,1])$ by covering $[0,1]$ with the set $(-\epsilon,1+\epsilon)$, which has measure $1+2\epsilon$.  Then the greatest lower bound of the set $\{1+2\epsilon \bigm| \epsilon > 0\}$ is just $1$ (though $S$ might be a little bigger than that set, so you really need to check all covers).
