TITLE: Why define measures on $\sigma$-rings?
QUESTION [8 upvotes]: I have the impression that modern texts deal almost excusively with measures on $\sigma$-algebras, while older texts, such as the one of Halmos, deal mainly with measures defined on $\sigma$-rings. I'm curious what motivated this change and in what context are $\sigma$-rings more natural domains for measures?

REPLY [3 votes]: Being unfamiliar with the older text, I can only speculate. One explanation is that one prefers to work with sets of $\sigma$-finite measure: those that can be written as a countable union of sets of finite measure. For example, sets of $\sigma$-finite length ($1$-dimensional Hausdorff measure) in the plane form a $\sigma$-ring, not a $\sigma$-algebra. It is rather fruitless to think about 1-dimensional measure of the complement of a line, so removing such sets from consideration seems reasonable.