Borel sigma-algebra
Note:
- Not to be confused with the Borel sigma-algebra generated by which, for a given topology (X,O) is denoted B(X,J):=σ(O) or just B(X) if the topology is implicit.
- (This page) The Borel σ-algebra refers to B(R), with it's usual topology (the topology induced by the absolute value metric, |⋅|).
- Because it is so common we simply denote it B
- Again, because it is so common, rather than saying a map is A/B-measurable, we may just say it is A-measurable
Definition
The Borel σ-algebra is a[Note 1] σ-algebra on R[1]. It is generated by the open sets of the metric space (R,|⋅|). We denote it as:
- B:=σ(O) where O denotes the open sets of (R,|⋅|)[Note 2]
This is actually a special case of the Borel σ-algebra generated by, rather than writing B(R) we simply write B
The Borel σ-algebra can also be defined on Rn, that is done as follows:
- Bn:=B(Rn)[1] with the usual topology on Rn (the metric given by the Euclidean norm will do)
Again, this is a special case of the Borel σ-algebra generated by a topology; this time it is the metric space (Rn,|⋅|).
Generators
There are many generators of Bn (just use n=1 for B itself) - some are listed here. First here are some non-obvious definitions:
- [[a,b))⊂Rnmeans a and b are n-tuples that denote the half-open-half-closed rectangles:
- [[a,b)):=[a1,b1)×[a2,b2)×⋯×[an,bn)⊂Rnwith the convention of:
- [ai,bi)=∅if bi≤ai and of course
- [[a,b))=∅if any of the [ai,bi)=∅ - this is trivial to show.
- [ai,bi)=∅
- The notation of ((a,b)):=(a1,b1)×(a2,b2)×⋯×(an,bn)is similarly defined.
- [[a,b)):=[a1,b1)×[a2,b2)×⋯×[an,bn)⊂Rn
- J∘rat:={((a,b))| a,b∈Qn}
- Jrat:={[[a,b))| a,b∈Qn}
- J∘:={((a,b))| a,b∈Rn}
- J:={[[a,b))| a,b∈Rn}
Claim | Proof route | Comment |
---|---|---|
Bn:=σ(O) - open[1] | Trivial (by definition) | |
Bn=σ(C) - closed[1] | Showing σ(C)⊆σ(O) and σ(O)⊆σ(C) - see Claim 1 | This is true for any Borel σ-algebra generated by a topology |
Bn=σ(K) - compact[1] |
TODO: There's quite a few steps and theorems required (eg: compact set in Hausdorff space is closed) |
Link with generated borel sigma algebra - which requires a Hausdorff metric space I believe |
Bn=σ(O)=σ(J∘rat)=σ(J∘) | Claim 2 | |
TODO: Check this and the method from the book - page 67 of my notes |
Proof of claims
Claim 1: σ(O)=σ(C)
Claim 2: Todo - even write this
Notes
- Jump up ↑ There are of course others, for example P(R) is always a σ-algebra but is much larger than the Borel one
- Jump up ↑ Conventionally, J denotes the open sets, but in measure theory this seems to denote the sets of half-open-half-closed rectangles, and it is too common to ignore
References
- ↑ Jump up to: 1.0 1.1 1.2 1.3 1.4 Measures, Integrals and Martingales - Rene L. Schilling