Generated and Borel sigma-algebras
Section of Probability spaces and events, part of Stochastic processes
Generated and Borel sigma-algebras
| Type | Section |
| Taught in | Probability spaces and events |
| Part | Stochastic processes |
The sigma-algebra generated by a collection \mathcal{C}, written \sigma(\mathcal{C}), is the smallest sigma-algebra containing \mathcal{C}. For a single set A it is \{\emptyset, A, A^c, \Omega\}, with duplicates removed if A is empty or all of \Omega. For two sets A and B, the four atoms
A\cap B,\qquad A\cap B^c,\qquad A^c\cap B,\qquad A^c\cap B^c
form a partition of \Omega. Every set in \sigma(A,B) is a union of some of these atoms.
On \mathbb{R}, the Borel sigma-algebra \mathcal{B}(\mathbb{R}) is the sigma-algebra generated by the open intervals. It contains the closed intervals, the half-open intervals, and every countable set, because each singleton is the intersection of a shrinking sequence of open intervals. Calling a set a Borel event says that it belongs to this sigma-algebra. It does not say that every subset of \mathbb{R} is Borel.
Used in
See also
Probability space · Borel–Cantelli lemmas · Limits of sequences of events · Sigma-algebra · Countable additivity