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Let's take the category of measure spaces with objects $(X,\mathcal{F},\mu)$ and avoid the morphisms for now (I'm not sure what they should be), where $X$ is a set, $\mathcal{F}$ is a $\sigma$-algebra, and $\mu$ is a measure on this space. If we forget that $\mathcal{F}$ is a $\sigma$-algebra and just think of it as an algebra, we have a forgetful functor into the category of premeasures.

There is the usual way of getting a measure from a premeasure by using outer measures and Caratheodory's criterion. On the other hand, there is usually an adjoint to a forgetful functor.

Is there some way to fill in the details (pick out morphisms "correctly" maybe) so that these ideas are the same thing?

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    $\begingroup$ What about the measurable and measure preserving maps as morphisms? The Caratheodory construction does not really give back the the measure space $(X,\mathscr F,\mu)$ but its completion. $\endgroup$
    – Jochen Wengenroth
    Commented Jan 11, 2015 at 11:01
  • $\begingroup$ That was a natural choice, or measurable maps that don't increase measure. It is okay if the Caratheodory construction gives back the completion. Going through a forgetful functor and back through its left adjoint is rarely the identity, and through a right adjoint doesn't have to be either. $\endgroup$
    – GraduateStudent
    Commented Jan 12, 2015 at 15:56

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