When defining a measure for a set we usually cannot hope to make every subset of measurable. Instead we must usually restrict our attention to a specific collection of subsets of , requiring that this collection be closed under operations that we would expect to preserve measurability. A -algebra is such a collection.
It follows from the definition that any -algebra in also satisfies the properties:
Any intersection of countably many elements of is an element of .
Given any collection of subsets of , the -algebra generated by is defined to be the smallest -algebra in such that . This is well-defined, as the intersection of any non-empty collection of -algebras in is also a -algebra in .
For any set , the power set is a -algebra in , as is the set .
A more interesting example is the Borel -algebra (http://planetmath.org/BorelSigmaAlgebra) in , which is the -algebra generated by the open subsets of , or, equivalently, the -algebra generated by the compact subsets of .
|Date of creation||2013-03-22 12:00:28|
|Last modified on||2013-03-22 12:00:28|
|Last modified by||yark (2760)|