What is the relation between semirings in measure theory and semirings in abstract algebra?
Why are they called the same?
You can see : http://en.wikipedia.org/wiki/Semiring
What is the relation between semirings in measure theory and semirings in abstract algebra?
Why are they called the same?
You can see : http://en.wikipedia.org/wiki/Semiring
They are different. The name for the object in measure theory is inspired by the object in algebra, viewing $\cap$ as multiplication, $\cup$ as addition, $\emptyset$ as $0$ and $1$ is the union of all sets in $S$ (if this is an element of $S$ - the definition doesn't require that it is).
The difference is that it isn't necessarily closed under those operations, instead we have the condition that, if $A\in S$ and $B\in S$ then there exists a finite number of mutually disjoint sets $C_i \in S$ such that $A\setminus B = \cup_{i=1}^n C_i$.
The name in algebra comes from the fact that a semiring is a weakened form of a ring in which addition is an abelian semigroup rather than an abelian group.
The name in measure theory comes from looking at a generalization of a ring of sets.