What are they. I mean Really.

http://en.wikipedia.org/wiki/Borel_set

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In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel.

For a topological space X, the collection of all Borel sets on X forms a σ-algebra, known as the Borel algebra or Borel σ-algebra. The Borel algebra on X is the smallest σ-algebra containing all open sets (or, equivalently, all closed sets).

Borel sets are important in measure theory, since any measure defined on the open sets of a space, or on the closed sets of a space, must also be defined on all Borel sets of that space. Any measure defined on the Borel sets is called a Borel measure. Borel sets and the associated Borel hierarchy also play a fundamental role in descriptive set theory.

In some contexts, Borel sets are defined to be generated by the compact sets of the topological space, rather than the open sets. The two definitions are equivalent for many well-behaved spaces, including all Hausdorff σ-compact spaces, but can be different in more pathological spaces.

I'm caught up on what a countable union is and how the set can be countable if it's a union of open sets. I thought open sets on R, or closed sets for that matter, were not countable?

Also what are they good for?

A countable union means a union of a countable number of sets, not that any of the sets necessarily have a countable number of elements. You are correct that open sets in R are not countable, in that they have an uncountable number of elements.

Saying a set is Borel set simply means that it is not too weird. It is what mathematicians call a well-behavedness condition. In order to prove theorems about certain objects, you usually have to make a number of assumptions that they are not too weird ('badly behaved').

A Borel set is a subset of the n-dimensional Euclidean space R^n. For the one-dimensional space R, most subsets you come across will be Borel. An example of a non-Borel set is the set of all irrational numbers between 0 and 1. That is a very naughty set and you can scarcely do anything with it.

Naughty and nice. I like that. For the set of all irrational numbers between 0 and 1, would that exclude 1/2? It would have little pin-prick holes in it? Would the same between 1 and 2 be naughty or nice?

So if I have three segments on the real line say: (2,5), [4,7]. and [6,9) the union would be a Borel set? Open or closed sets don't matter. Continuous does?

Any examples of truly naughty behavior?

andrewk wrote:A countable union means a union of a countable number of sets, not that any of the sets necessarily have a countable number of elements.

Thank you so much for this. The definitions I read did not make that distinction clear and that is where my mind turned mule on me.

A countable union is the union of countably many sets. The sets themselved do not have to be countable.
A σ-Algebra A on a sample space Ω is a subset of the Power set of Ω fulfills the following criteria:
Ω is in A
if A in A then Ω/A is in A
if An is in A for all n in N, then the union of all the An is in A
The last one gives countable unions.

We can define a σ-Algebra σ(B) that is generated by a subset B of the power set of Ω as the set of all sets that you get by using the above rules, or equivalently as the intersection of all σ-Algebras on Ω that contain B.
The Borel σ-Algebra on R then is σ({[a,b], a,b in R}) or σ({]a,b[, a,b in R}), or σ({[a,infinity[, a in R})

Now, the main point of having a σ-Algebra is the construction of measure spaces, i.e. triplets (Ω, A, µ) where µ is the measure. If Ω is countable, you don´t have any issues if you pick A as P(Ω), but if Ω isn´t countable, then there are some sets in P(Ω) for which you can´t consistenly define a measure. So you would like to have a σ-Algebra that excludes these sets, but includes as many others as possible. The Borell-Algebra does fit this bill - it includes all intervals, all countable subsets, all countable unions of intervals and all unions of countable subsets with countable unions of intervals and of course it includes Ω and {}.

To give an example for a set that is in P(R) but not in B(R):
1. Start by defining sets Di={i+q| q in Q} for an index set I, so that the union of all Di is R and that Di=Dj iff i=j (you have broken R into uncountably many countable dijunct subsets)
2. Now use the Axiom of choice to construct a set G taking one element from each D.
3. Then define G(q)={q+g|g in G} for q in Q. (you have now broken up R into countably many uncountable disjunct subsets).

(I´m somewhat sketchy on this, it´s been a while since I did this, so I might be off somewhat).

It turns out that the sets G(q) break many measures we would like to use (probability measures, lebesque measures). But in step 2 we constructed a union of uncountably many sets. So these sets are in P(R), but not in B(R) and that´s why we use the latter when defining measures on uncountable sets.

The set of all irrational numbers between 0 and 1 is Borell:
Construction:
You easily get single element sets from the Borell σ-Algebra (noting that open and half open intervals both define B(R), we can note that for a,b in R [a,b[ and ]a,b[ are in B(R), and this implies {a} in B(R) for a in R) , including the sets Sq={q}, q in Q.
Then we take a countable union of the sets Sq and get Q.
[0,1] is in B(R)
The union between ]-infinity,0[, ]1,infinity] and Q is in B(R)
Hence the set of all irrational numbers between 0 and 1 is in B(R).

Actually SOS, I forgot just how antisocial Borel sets were. The set of irrationals between 0 and 1 is pretty bad, but still Borel, contrary to my over-hasty suggestion above. The Cantor set, which is even weirder than the irrational one, is also Borel. The wikipedia article on Borel sets gives only one constructive example of a non-Borel set. Non-Borel sets are so dangerous they need to be locked up in padded rooms and kept under constant armed guard.

It's usually reasonable to interpret any mathematical paragraph that starts by saying 'consider the Borel sets in Rn' as meaning 'consider all the subsets of Rn that you are ever likely to think of'.

Shit. I might have had some of those sets get loose on my doodle pad. Thanks for the warning!

So what is the essence of a non-borel then if not well-behavedness?

Non-Borel generally wreaks havoc with measures (i.e. it´s not well-behaved for a particular type of thing we´d like to do). For instance I can figure out the probability that a urianium nucleus will decay in a borel set with no issues. But If I wanted to know the probability of it decaying in a number of seconds that is in G as defined above the answer wouldn´t make sense: It´d be 0, 1, 36 and infinity, all at once. If I take a pice of string I can figure out its lenght and say that it´s lenght is 1. If I took out all the rational numbers the remainder of the string would still have lengt 1. If I took out all the numbers that are also in G it would have any lenght. These sets are not behaving well for measures.

The article on non-measurable sets has a construction similar to the one I gave above. It also notes that there´s a proof that all constructions of non-measurable sets require applying the AOC to an uncountable set of sets, which is rather cool.

I do see issues with your final statement: You can´t define a uniform distribution on P(R) and I have the suspicion that the R(R)/B(R) (the set of Non-Borel sets on R) is not a Borel set on P(R), which would imply that you can´t even define a probability if you restrict your distribution to some subset of P(R).

Edit: IIRC a simple combination of general relativity with quantum mechanics gives you infinite probabilities. This implies that non-Borel sets crop up there somewhere, because Borel sets don´t allow this to happen.

I'm catching a glimpse with the measurability. Cool shit. I'm a noob to set theory. About a month or two in part-time.

The Banach-Tarski paradox involves using the Axiom of Choice to disassemble a sphere into a finite number of non-overlapping pieces, which can then be put back together in a different way to yield two identical copies of the original ball - ie you double the volume by taking it apart and putting it together again.

Although I don't know for sure, I would not be at all surprised to find that those pesky non-Borel sets are involved in what is clearly an illegal and immoral activity.

Most non-Borel sets discussed appear to require the Axiom of Choice for their construction, and not everybody accepts that axiom. However the wikipedia article on Borel sets does include a Borel set that doesn't require the axiom. If not for that, I might have to conclude that there's not necessarily any such thing as a non-Borel set.

Banach-Tarski paradox. Pure evil. Some kindling and a stake!

I Beg Le question and eventually it all comes together. Glad I asked about Borel sets. It was almost random but turns out to be the key.

http://en.wikipedia.org/wiki/Measure_theory

This is all about measure and probability. The thread that sourced the question was causality.

I have this idea about how to teach things that is straight out of neuroscience. It nauseates me that the schools have not figured this shit out yet. When you build a house you do not start by hanging the siding and putting in the windows. You need a frame. IF someone knows in general what the shape of the frame is then the rest of the project will be a little bit easier.

So what can be said about Borel sets and measures and sigma-algebras that a new student would grasp immediately? What is the overall frame of the thing? If you look at the wiki link here you see a picture of an area you can measure.

http://en.wikipedia.org/wiki/Lebesgue_integration

That is a piece of frame that must go up first. Saying this: "Lebesgue integration" to someone is not so useful without the picture. Makes you sound pretty smart though. I'll give it that.

SpeedOfSound wrote:So what can be said about Borel sets and measures and sigma-algebras that a new student would grasp immediately? What is the overall frame of the thing? If you look at the wiki link here you see a picture of an area you can measure.

Well, there are the definitions. The main issue here is that all of this stuff is the foundation of measure theory and the motivation for it comes from the things that arise later. Historically this foundation has been changed when problems in the building appeared. If you look at Pascals wager for instance, you´ll see a case of using probability theory at a time when the foundation wasn´t that clear (and as it turns out the current foundations don´t allow it any more - the wager is based on deriving an expected value for a random variable that has no expected value). So a lot of the things in these foundations only really make sense when you understand the stuff that follows and realize that not making particular statements early one would really get you into trouble in that proof over here or this definition, which stops being unique. When I went to my first probability theory course I thought it´d be about probability, but that notion only got introduced after months of problems of the type "construct a non-Borel set", "let D be a subset of P(M) that is dynkin and closed under finite intersection: Show that D is a σ-algebra" and "Construct all σ-algebras on the set {1,2,3,4}". Later you understand why this is important, when you just do it it does seem rather pointless.

SpeedOfSound wrote:That is a piece of frame that must go up first. Saying this: "Lebesgue integration" to someone is not so useful without the picture. Makes you sound pretty smart though. I'll give it that.

Well, the image there doesn´t really show the crucial bits: How Lebesque differs from Riemann (which is the type of integration you learn in schools). The key advantages of Lebesque are the types of sets you can measure (the sets that are Riemann measurable are a true subset of the sets that are Lebesque measurable, and the two measures agree on them) and some theorems that apply to the Lebesque measure: in particular monotone convergence and Fatou's Lemma.

So, the best one can say is: It´s stuff you need to do before you get to the things you want to do and why you need to do it in the first place only becomes clear when you start doing the later stuff...

susu.exp wrote:
So, the best one can say is: It´s stuff you need to do before you get to the things you want to do and why you need to do it in the first place only becomes clear when you start doing the later stuff...

Which would be fine if our brains were computers but they are not. I think it an art to teach by building the frame first. No one said it would be easy or that everyone will be good at this art.

Two things are needed for effective learning. The first is a frame upon which to hang associations. The number of associations will increase the strength of the concept and hence memory and understanding. The second is an emotional weight to the memory. Without it you get no repetition and you get no Ca+2 and then you get no signals to increase synaptic strength. (gross simplification there and grievous errors but you get the picture)

An emotional weight is given by setting the goal. Where are we going with this thing? The association frame is given by same.

To write a really good piece on math (or even good fiction) is to create opening paragraphs that set this frame and intent intuitively and then start to drill down to the details.

Some of the wiki articles do this very well and others are total fail.

BTW. Thanks for your very good explanations and andrewk too. I have my frame now and I know where to start looking for the sheet rock.

I got J. Pearl Causality last week. Just bought this: First Look at Rigorous Probability Theory and this :First Look at Rigorous Probability Theory.

I am looking for something serious that I am not quite ready to describe. It would only confuse me if I tried. I will know it when I see it. Has to do with neurons and dynamical systems and transformations and statistics. It is to be a model for how we think. What happens to us when an environment makes some move on us. It's a bridge theory. Or some fucking thing. Right now it's just another wet dream.

What else should I buy in measure and probability and order theory?

SOS this probably doesn't relate directly to the direction you seek but, given your background in options valuation, you might find the following book interesting as an amusing sidebar. It develops the principles of option pricing in a more mathematically rigorous way than I have seen in other finance texts (eg Hull's book is the 'bible' of financial quants and an excellent general purpose text but it skates over the difficult maths). It touches on a number of the key aspects of measure and probability theory that underpin the development of pricing theory, such as the Radon-Nikodym derivative, the Cameron-Martin-Girsanov theorem and the Martingale Representation Theorem. Like many texts it gets a bit hard going at times and often omits a number of crucial steps, leaving the conscientious reader the chore of trying to prove that line n+1 follows from line n, but it's better than any other text I've seen that deals with this subject.

The book is 'Financial Calculus' by Baxter and Rennie.

I sometimes get my sigma-algebras mixed up with my topologies, because their definitions are very similar, and I studied topology long before I studied probability theory. Sometimes a sigma-algebra is the same as a topology, but usually they are different because:
- a sigma-algebra is required to be closed under complementation and countable intersections, whereas a topology is not
- a topology is required to be closed under any union, whereas a sigma-algebra is only required to be closed under countable unions.
So, despite the similarity, not every topology is a sigma-algebra, and not every sigma algebra is a topology.

The set of irrational numbers in (0,1) is not in the topology generated by the collection of all open intervals, because you would need a countably infinite intersection of open intervals to construct it from the base collection. However it is in the sigma algebra generated by the collection of open intervals, which is the collection of Borel sets over R.

susu.exp wrote:When I went to my first probability theory course I thought it´d be about probability, but that notion only got introduced after months of problems of the type "construct a non-Borel set", "let D be a subset of P(M) that is dynkin and closed under finite intersection: Show that D is a σ-algebra" and "Construct all σ-algebras on the set {1,2,3,4}". Later you understand why this is important, when you just do it it does seem rather pointless.

Do you have the name of the textbook that puts you through this prelim before it gets into probability?

SpeedOfSound wrote:Do you have the name of the textbook that puts you through this prelim before it gets into probability?

I´ve used a german textbook mostly for reference (which suffers from the n to n+1 issue adrew has alluded to) and the script the professor I took the courses at put out (german as well).
So, I´d look for lecture notes:
http://users.jyu.fi/~geiss/scripts.html (seems nice, though it doesn´t cover everything later on, a proof of the CLT in some version would have been nice for instance).