Generating the Borel (sigma-)Algebra

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Discussion Overview

The discussion revolves around the generation of the Borel sigma-algebra on the real numbers from a specific set F, which consists of finite unions of sets of the form (a,b], (-∞, b], (-∞,∞), and (a,∞). Participants explore the necessary steps to demonstrate that this set generates the Borel algebra and discuss the implications of this generation process.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant suggests that to show F is contained in the Borel algebra, it is necessary to demonstrate that all open sets can be formed from finite unions and intersections of the sets in F.
  • Another participant proposes that the converse must also be shown: that all sets of the form (a,b], (-∞, b], (-∞,∞), and (a,∞) can be formed from finite unions and intersections of open sets.
  • A participant expresses confidence in forming open sets from the sets in F but struggles with forming the sets (a,b] and (-∞, b] from open sets, considering the use of unions of (a, b + 1/n) as a potential method.
  • There is a clarification that both parts of the argument require countable unions and intersections, not just finite ones, as they are working within a sigma-algebra context.

Areas of Agreement / Disagreement

Participants generally agree on the steps needed to show the relationship between the set F and the Borel algebra, but there remains uncertainty regarding the specific methods to construct certain sets from open sets. The discussion is ongoing, with no consensus reached on the best approach.

Contextual Notes

Participants acknowledge the need for countable unions and intersections in their arguments, which may affect the validity of their proposed methods. The discussion highlights the complexity of the relationships between different types of sets within the framework of sigma-algebras.

matt.qmar
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Hi!

We're given a set F, and F is the set of all finite unions of sets of the form (a,b], (-∞, b], (-∞,∞), and (a,∞), for real numbers a & b.

I am trying to show that the sigma-algebra generated by F is actually the Borel algebra on the Reals.

(recalling that the Borel algebra of the reals is the sigma algebra generated by all the open subsets of the real line)

So I think there are two things to show - firstly, that we can generate the Boreal Algebra from these finite unions of sets of the form describe above (F is contained in the Borel algbera)

and secondly, that the Boreal algebra is the "smallest" such algebra,
and by "smallest", we mean that if F is contained in any other sigma algebra, say C, then the Borel Algebra is also conatined in C.

I think once we've shown those things, then we're OK. I'm just not entirely sure on where to start to go about showing them!

Any help much appriciated. Thanks!
 
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You need to show two things:

1) You can form all open sets by forming finite unions/intersections of sets of the form (a,b], (-∞, b], (-∞,∞), and (a,∞).

and conversely,

2) All sets of the form (a,b], (-∞, b], (-∞,∞), and (a,∞) can be formed by forming finite unions/intersections of open sets.
 
Thanks for the reply!

So I can see how to get the open sets from unions and intersections of those sets (Part 1)

But I'm not sure how to get all those sets from finitly many open sets (Part 2)... I can see how to grab the last two, but I'm not sure to form (a,b] and (-∞, b] from open sets; is there a trick by taking unions of (a, b + 1/n) or something along those lines? I know that is a common trick to catch closed endpoints... but I think that only works if we have an infinite union...

Thanks again!
 
matt.qmar said:
Thanks for the reply!

So I can see how to get the open sets from unions and intersections of those sets (Part 1)

But I'm not sure how to get all those sets from finitly many open sets (Part 2)... I can see how to grab the last two, but I'm not sure to form (a,b] and (-∞, b] from open sets; is there a trick by taking unions of (a, b + 1/n) or something along those lines? I know that is a common trick to catch closed endpoints... but I think that only works if we have an infinite union...

Thanks again!

Oh, I'm so sorry. I meant countable unions/intersections in both 1 and 2.
 
Oh, right. Of course! We have a sigma algebra. Thanks again to clarifying exactly what conditions allow that the sigma algebra is generated by a set.
 

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