Reference: Measure Theory by Halmos.

An example of a pathology: If measures are not σ-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 σ-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∞ is easier when the whole space is in the domain of the measure.

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.