Borel sigma-algebra

From Optimal Transport Wiki
Jump to navigation Jump to search

If X is any metric space, or more generally any topological space, the -algebra generated by the family of open sets in X (or, equivalently, by the family of closed sets in X) is called the Borel -algebra on X and is denoted by . Its members are called Borel sets. thus includes open sets, closed sets, countable intersections of open sets, countable unions of closed sets, and so forth. [1]

The Borel -algebra on is denoted by .


is generated by each of the following:

  • a. the open intervals: ,
  • b. the closed intervals: ,
  • c. the half-open intervals: , or ,
  • d. the open rays: or ,
  • e. the closed rays: or ,

Product Borel -algebra

Let be metric spaces and let , equipped with the produc metric. Then . If the 's are separable, then .

Proof. By proposition in product -algebra, is generated by the sets , where is open in . Since these sets are open in , . Suppose is a countable dense set in , and let be the collection of balls in with rational radius and center in . Then every open set in is a union of members of , which is countable. Moreover, the set of points in X whose jth coordinate is in for all j is a countable dense subset of X, and the balls of radius r in X are merely products of balls of radius r in the 's. It follows that is generated by and is generated by . Therefore, .


  1. Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, Second Edition, §1.2