Understanding Borel Sets: A Beginner's Guide

[datatec]

Hi guys!

Could somebody please explain to me in the most basic of ways what a Borel set is...

Thanks

 

[matt grime]

Consider the sets [a,b) for any a and b in the reals (and also allow b to be infinity).

A borel set is then something that may be obtained by repeatedly using the operations if union, intersection and complement to these sets and any sets that we obtain in the process too.

ok, sounds hand wavy and uninformative. sorry. this is called expressing it in terms of a "basis".

It is easier for me to put it this way: essentially every subset of the real numbers that you an describe is a Borel set. I hope I don't saty something false here, but the only way you can define a subset of the reals that is not a borel set is by using the axiom of choice, and we can perhaps think of this as being "pathologically" bad and not a representation of any set you'll meet in "real life".

here is a link showing just how hard it is to define a nonmeasurable set.

http://www.ma.ic.ac.uk/~boz/M3P2/Non-MeasSet/non-meas.html

 

[mathman]

Def: sigma field is a collection of sets closed under countable unions and countable intersections and complements.

Def: Borel field is smallest sigma field containing all open sets. Borel sets are sets within Borel field.

For real line using usual topology (open sets defined from open intervals), there are sets which are NOT Borel sets, although you need to use the axiom of choice to prove they exist.