Background: I am currently I am reading Angus and Taylor's, "General Theory of Functions and Integration". I'm reading it because they do a pretty nice job of explaining things but I have also read other books that discuss the Lebesgue integral and define outer measure in the same manner. I've always been confused by the definition. Angus and Taylor define the outer measure as:
$u^{*}(E) = glb\left(\sum_{n} |I_{n}| : \space\{I_{n}\} \text{ is a Lebesgue covering of E} \right )$.
What I don't understand is, given that one might be able to construct some gigantic covering, how can there be any bound on $u^{*}(E)$ ?
Maybe if someone could provide an example where the infimum is actually used, that might open my eyes. For example, after they define outer measure, they say that:
"a single open interval I is a Lebesgue covering of itself. Therefore, $u^{*}(I) \le |I|$".
I don't see why that is the case because I don't see how the infimum comes into play ?
Thanks a lot (in advance) for any example or statement that could help me to understand the definition. I have read the following thread - which may be of some relevance here, but this does not answer the particular issues I've raised above (at least not directly).