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?