What is a Borel Set? Explained with Examples

[hassman]

Hi...

I have searched but the explanations that are given are too abstract. Why is it so difficult to use an example to show what a Borel set is?

Assume X = {1, 2, 3}. Then the power set of X is a topology. Borel set is defined on topologies right?

So what would be then a Borel set?

Or perhaps someone could explain to me without the above example, but in clearer terms what a Borel set is?

 

[Fredrik]

The abstract definition is "the smallest σ-algebra on X that contains all the open subsets of X". So in this case, since a σ-algebra on X is a subset of the power set of X, and since the set of all open subsets of X is the power set (by your choice), the set of Borel sets is the power set. With this topology, a set is Borel if and only if it's open.

I suggest you look at exercises 1.9.3 and 1.9.8 in this book, and that you also read the comments after the exercises.

 

[g_edgar]

So, as noted, the Borel sets for a finite (or even countable) metric space turn out to be *all* subsets, so there is no need for a separate study of them. The best place to investigate Borel sets is the metric space of the real line. In that case, NOT every set is a Borel set.

 

[hassman]

thank you. i will buy that book probably. looks pretty good. i cannot make the exercises though as i lack the fundamentals.

 

[JSuarez]

So, as noted, the Borel sets for a finite (or even countable) metric space turn out to be *all* subsets

No, this is true only for the trivial (full) topology. For an extreme couterexample, consider the other trivial topology \left\{\emptyset,X\right\}.

 

[Hurkyl]

No, this is true only for the trivial (full) topology. For an extreme couterexample, consider the other trivial topology \left\{\emptyset,X\right\}.

That cannot be the topology of a metric space, though, unless X is empty or a single point.

 

[RoNN|3]

It's been some time since I last saw them, so someone correct me if I am wrong in thinking that these are precisely the Borel sets:
- open sets are Borel
- any set obtained from other Borel sets via unions and intersections is Borel

 

[g_edgar]

It's been some time since I last saw them, so someone correct me if I am wrong in thinking that these are precisely the Borel sets:
- open sets are Borel
- any set obtained from other Borel sets via unions and intersections is Borel

complements, too

countable intersections

countable unions

but keep repeating these operations (finitely many repetitions may not be enough...)

 

[RoNN|3]

Indeed, I forgot the complements.