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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?
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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.

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