Definition of B1

Definition: B¹ denotes the Borel sigma-algebra of the real line. It will contain every open interval by definition, which implies that it contains every closed interval and every countable union of open, half-open, and closed intervals.

What won't it contain? In practice, only obscure sets. Here is an example: Define the equivalence class ~ on the real line such that x~y (read: x is in the same equivalence class as y) if x-y is a rational number. Now consider the set of all numbers in [0,1] such that none of them are in the same equivalence class. How many members of that set are there? Well, it's not a countable number. This set is not in B¹.

Terms related to B(1):

• Borel sigma-algebra
• Borel Set
• Transition Economics

About.Com Resources on B(1):

• What is Real Analysis?
• Books to Study Before Going to Graduate School in Economics
• Math Camp for Economics Graduate Students

Writing a Term Paper? Here are a few starting points for research on B(1):

Journal Articles on B(1):

• Existence of Optimal Stochastic Control Laws
• Survey of Measurable Selection Theorems
• Valuation Equilibrium with Clubs