202405211412
Status: #idea
Tags: Measure Theory, Probability Theory
Borel Sets
A Borel set is defined by its appartenance to a Borel-Sigma Algebras. They are also called Borel measurable sets to put emphasis on that fact.
They are subsets ([Events]) to which we assign probabilities in Continuous Probability Spaces (actually in Probability Spaces in general.) Interestingly it is really difficult to define a set that is not a Borel set. Such sets exist, but you must go out of your way to define them after all if a set can be represented as the countable unions or intersections of open intervals or closed intervals, it is Borel.
For that reason such sets are ideal to use in probability theory and measure theory because basically any set we could ever conceivably want to deal with will be a Borel set (therefore we can define a measure on it), but all the nasty stuff is kept out.
How to Argue Something Is Borel:
- [Show that the Complement of the Set is Borel]
- Represent the Set as an Intersection of Open Sets