What is the difference between $\sigma$-algebra, $\sigma$-ring, and field of sets? I don't really understand the difference between these stuff. They look really similar. What is the difference between those? Which one do people use in measure theory, and probability related things and why? Is there any advantage to use one over the other?
 A: A $\sigma$-algebra of subsets of $X$ is a $\sigma$-ring of subsets of $X$ which contains $X$.
Unlike $\sigma$-algebra and $\sigma$-rings, field of sets are not closed under countable intersections or unions, but they contain $X$. So, $\sigma$-algebras are fields of sets (also called $\sigma$-fields), but a $\sigma$-ring might not contain $X$ and hence might not be a field of set. '$\sigma$' is generally used to emphasize countably infinite properties.
$\sigma$-algebras are used in measure theory since they model the set of measurable subsets of a measure space. 
Asking if one of these has an advantage over the others is a little strange, but I suppose you could say a field of sets has the advantage of being the most general, while the $\sigma$-algebra is the most specific (if you can call these advantages.) Each of them can be used in any situation where the axioms are suited.
A: I think examples will make a clarification between σ-ring and σ-algebra. 
Take A as the collection of all countable,which includes finite sets,subsets of R and B as all subsets of R which is countable or its complement is countable.
A is σ-ring which is not a σ-algebra but latter is a σ-algebra....
And adding the set X itself to a σ-ring makes the difference because for now it should be closed under complements for being a σ-ring it should be closed under relative complements.
Hence σ-algebra is the σ-ring containing X.
