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.