On Borel sets of the extended reals

In summary, "On Borel sets of the extended reals" discusses the properties and characteristics of Borel sets within the context of the extended real number system, which includes both positive and negative infinity. The paper explores the construction of Borel sets through iterative processes involving open and closed intervals, highlights their significance in measure theory and integration, and examines their applications in probability and real analysis. The work also addresses the challenges posed by the inclusion of infinity in the topology and the implications for various mathematical operations and functions defined on these sets.
  • #1
psie
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TL;DR Summary
I am reading Folland's text on real analysis. He defines the Borel -algebra on the extended reals and says this coincides with the usual definition of the Borel -algebra being generated by open sets. I don't see how they coincide.
On page 45 in Folland's text on real analysis, he writes that we define Borel sets in by . Then he remarks that this coincides with the usual definition of the Borel -algebra if we make into a metric space with metric .

I don't see how the two coincide and would be grateful if someone could explain in more detail. I see how is a metric on , where e.g. should evaluate to , but I don't see how coincides with the usual definition of being generated by the open sets in . In other words, how to show
 
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  • #2
Ok, here's an attempt at a solution:

: If is open in then by definition of subspace topology, is an open set in , thus belongs to . We've shown , hence also the -algebra generated by is a subset of .

: Let . Then and is a Borel set of . Now, since , the Borel -algebra on is the restriction of the Borel -algebra of to , i.e. . Because , there's a Borel set in such that , and since ( is open in , hence Borel in ), is a Borel set in . also belongs to , since it is finite (it is an intersection of singleton, i.e. closed sets, and hence Borel).
 
  • #3
By usual definition means that is a Borel set formed from open sets in through the operations of countable union, countable intersection (countable intersection can be got from countable union and relative complement (De Morgan law) and it can be skipped) and relative complement. Every open set in is in too because of . This means that is a set formed from open sets (the same open sets as in the first sentence, so open rays are not included ) in through the operations of countable union, countable intersection and relative complement. After including open rays into the open sets it can be said that is a set formed from the open sets in through the operations of countable union, countable intersection and relative complement.
Every set can be equal to or can be formed by countable union of and open rays.
 
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