In measure theory you have 'algebra's' and 'rings' as subsets of the powerset of the underlying set of the measurable space. If I am well informed then you speak of an algebra if it is closed under finite union and complement, and of a ring if complement is interchanged here by difference. My question:
Is there a connection between these concepts and the well-known algebra's and rings in algebraic theory?
In fact I don't think so, but - if I am right in this - then why are these terms used in this context?
An algebra in measure theory is a class, say '$\alpha$', closed under all finite set operations, you can have $\sigma$-algebras, which can be also called $\sigma$ rings as they hold connective laws. So yes to your question as long as you allow for countable unions only.