Borel sigma-algebra

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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))Rn
    means a and b are n-tuples that denote the half-open-half-closed rectangles:
    • [[a,b)):=[a1,b1)×[a2,b2)××[an,bn)Rn
      with the convention of:
      • [ai,bi)=
        if biai and of course
      • [[a,b))=
        if any of the [ai,bi)= - this is trivial to show.
    • The notation of ((a,b)):=(a1,b1)×(a2,b2)××(an,bn)
      is similarly defined.
  • Jrat:={((a,b))| a,bQn}
  • Jrat:={[[a,b))| a,bQn}
  • J:={((a,b))| a,bRn}
  • J:={[[a,b))| a,bRn}
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(Jrat)=σ(J) Claim 2

TODO: Check this and the method from the book - page 67 of my notes


Proof of claims

[Expand]

Claim 1: σ(O)=σ(C)

[Expand]

Claim 2: Todo - even write this

Notes

  1. Jump up There are of course others, for example P(R) is always a σ-algebra but is much larger than the Borel one
  2. 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

  1. Jump up to: 1.0 1.1 1.2 1.3 1.4 Measures, Integrals and Martingales - Rene L. Schilling