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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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# Generating the Borel (sigma-)Algebra

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