The following is said without further explanation in Folland's *Real Analysis*:



> 
> Some authors prefer to take the domains of measures to be $\sigma$-rings rather
>  than $\sigma$-algebras. The reason is that in dealing with "very large"
>  spaces one can avoid certain **pathologies** by not attempting to measure "very large"
>  sets. However, this point of view also has technical disadvantages, and it is no longer
>  much in favor.
> 
> 
> 


* Could anyone come up with some references where the authors "take the domains of measures to be $\sigma$-rings rather
than $\sigma$-algebras"?
* What examples of "pathologies" may Folland refers to?
* What "technical disadvantages" does "this point of view" have?



Why are $\sigma$-algebras preferable to $\sigma$-rings?
Reference: Measure Theory by Halmos.


An example of a pathology: If measures are not $\sigma$-finite, you have to be careful how to formulate Fubini's theorem. The old way of doing this was to have the measure defined on the $\sigma$-ring of those sets on which the measure is σ-finite. The current way of doing this is to assume that the measure is "locally determined". 


An example of a technical disadvantage: Defining spaces such as $L^\infty$ is easier when the whole space is in the domain of the measure.


