Let be a set with which satisfies the following properties
i.e. is a set of subsets of that contains the , and is closed under complement and countable union. A subset that satisfies and is called an algebra. When it satisfies , it satisfies some interesting properties.
By DeMorganās principle, it is trivial that is then closed under countable intersection. It it also clear that is also closed under finite union and intersection. Furthermore, as we may assert .
Note that we also have the following
thus is also closed under set difference.
For such a tuple , is called a measurable space, is called a -field while elements of are called measurable.
It is trivial that a -field exists for any , notably and . A less trivial example of a -field is the set of countable or co-countable sets, defined for some with an uncountable number of elements as
The relationship between and a certain subset can be expanded upon by inheritance and generation. A natural example of a -algebra in topological space is the set of Borel Sets.