Measurable function

From Optimal Transport Wiki
Jump to navigation Jump to search

Let and be measure spaces. A map is -measurable if for all This definition is analogous to the definition of a continuous function in a topological space, in which one requires that the preimage of each open set is open. In the case of the Lebesgue measure (or any Borel measure), all continuous functions are measurable.

Examples of measurable functions

  • A function is called a Lebesgue measurable function if is - measurable, where is the class of Lebesgue measurable sets and is the Borel -algebra on the extended real numbers.
  • A function is called Borel measurable if is -measurable.

Basic theorems of measurable functions

  • Let and be measure spaces. Suppose that is generated by a set . A map is -measurable if for all
  • Let , , and be measure spaces. If a map is -measurable and is -measurable, then is -measurable. In particular, if is Borel measurable and is Lebesgue measurable, then is Lebesgue measurable.
  • Let , , , and be measure spaces. If a map is -measurable and is -measurable, then is -measurable when

Properties of borel measurable functions

  • If is a measure space and measurable, then the following functions are measurable [1] :
    • where we define
    • where
    • where
    • , if the limit exists for all
  • If measurable, there exists a sequence (each a Simple Function ) satisfying p.w., i.e.

Non-measurable functions

It is possible to define functions that are not measurable. For example, if represents the Vitali set, then the indicator function is not Borel-measurable. This follows directly from the fact that , which is a non-measurable set, despite the fact that is closed in (and hence ).

  1. Craig, Katy. MATH 201A Lecture 11. UC Santa Barbara, Fall 2020.