Generating the Borel (sigma-)Algebra

[matt.qmar]

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!

 

[micromass]

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.

 

[matt.qmar]

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!

 

[micromass]

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.

 

[matt.qmar]

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.