measurable function
In other words, the inverse image of every -measurable set is -measurable. The space of all measurable functions is denoted as
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.
Example:
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Let be a subset of a measurable space . Then the characteristic function is a measurable function if and only if is measurable.
Title | measurable function |
Canonical name | MeasurableFunction |
Date of creation | 2013-03-22 12:50:50 |
Last modified on | 2013-03-22 12:50:50 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 18 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 28A20 |
Synonym | Borel measurable |
Related topic | ExampleOfFunctionNotLebesgueMeasurableWithMeasurableLevelSets |
Related topic | LusinsTheorem2 |
Related topic | BorelGroupoid |
Related topic | BorelMorphism |
Defines | Borel measurable function |