TITLE: Is there a similar concept for a sigma algebra like a base for a topology?
QUESTION [12 upvotes]: For both a sigma algebra and a topology, we can talk about their generators. 
For a topology, a base is a special generator only using union, which is a useful concept in topology. 
In parallel comparison, is there a similar concept for a sigma algebra like a base for a topology? How useful can it be? 
Thanks and regards!

REPLY [3 votes]: A semi-field $\mathcal{S}$ on $\Omega$ is a class of subsets of $\Omega$ such that:

$\Omega\in\mathcal{S}$
$A,B\in\mathcal{S}\Rightarrow A\cap B\in\mathcal{S}$
$A\in\mathcal{S}\Rightarrow A^c$ can be written as a finite disjoint union of sets in $\mathcal{S}$

The class of subsets of $\Omega$ formed by taking all finite (disjoint) unions of sets in $\mathcal{S}$ form a field, $\mathcal{F}$. If we take all possible countable unions of sets in 
$\mathcal{F}$, then we get a $\sigma$-field. 
Hence I think semifield in measure theory is an analogue of base in topology.