Perhaps I don’t speak English as well as I thought I did. In Folland, and other sources, I have encountered the following definitions: A “measurable space” consists of a set X together with a sigma-algebra M, which is a subset of the power set of X. And a “measurable set” is any element of that sigma-algebra.
In these definitions, there is nothing that says the measurable sets in question can be measured. Thus, for example, consider any set X that has a non-measurable set E which is a subset of its power set. Then the set X, together with the set E, its compliment, and the null set form a sigma-algebra, and thus a “measurable space.” Since E is an element of this sigma-algebra it is “measurable”, although by assumption it is non-measurable. A contradiction.
The definitions thus appear to be utter non-sense. But perhaps, I do not understand English.