This is related to my measure theory class, but it's not homework. The motivation behind this post is to understand the exact relationship between the three objects mentioned in the title.

[Carathéodory's theorem](http://en.wikipedia.org/wiki/Carath%C3%A9odory%27s_theorem_(measure_theory)) states that given an outer measure $\varphi$ on a space $X$, the set of $\varphi$-measurable sets form a $\sigma$-algebra and $\varphi$ restricted to the measurable sets is a countably additive complete measure. I am wondering if this admits a converse:

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> **Q1.** Given a complete measure space $(X, \Sigma, \mu)$, does there always exist an outer measure $\varphi$ satisfying the following conditions?
> 
> 
> 1. A set $E \subseteq X$ is $\varphi$-measurable iff $E \in \Sigma$,
> 2. $\varphi$ restricted to $\Sigma$ is $\mu$.
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> 
> 

A related question is:

> 
> **Q2.** Given a $\sigma$-algebra $\Sigma$ on $X$, can I always define a complete measure $\mu$ on the measurable space $(X, \Sigma)$?
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> 
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To pinpoint the source of my confusion, I feel that the word *measure* is used in several related, but seemingly different, ways; e.g., "measure space", "measurable space" and "outer measure". If the answers to both the questions are in the affirmative, then it would become clearer to me that these different usages are very related after all.