Any measurable function into , where is the Borel sigma algebra of the real numbers , is called a Borel measurable function.11More generally, a measurable function is called Borel measurable if the range space is a topological space with the sigma algebra generated by all open sets of . The space of all Borel measurable functions from a measurable space is denoted by .
Similarly, we write for , where is the Borel sigma algebra of , the set of extended real numbers.
Remark. If and are measurable functions, then so is , for if is -measurable, then is -measurable, and is -measurable. But , which implies that is a measurable function.
Let be a subset of a measurable space . Then the characteristic function is a measurable function if and only if is measurable.
|Date of creation||2013-03-22 12:50:50|
|Last modified on||2013-03-22 12:50:50|
|Last modified by||CWoo (3771)|
|Defines||Borel measurable function|