I'm studying Probability theory, but I can't fully understand what are Borel sets. In my understanding, an example would be if we have a line segment [0, 1], then a Borel set on this interval is a set of all intervals in [0, 1]. Am I wrong? I just need more examples.
Also I want to understand what is Borel $\sigma$-algebra.
To try and motivate the technical answers, I'm ploughing through this stuff myself, so, people, do correct me:
Imagine Arnold Schwarzenegger's height was recorded to infinite precision. Would you prefer to try and guess Arnie's exact height, or some interval containing it?
But what if there was a website for this game, which provided some pre-defined intervals? That could be quite annoying, if say, the bands offered were $[0,1m)$ and $[1m,\infty)$. I suspect most of us could improve on those.
Wouldn't it be better to be able to choose an arbitrary interval? That's what the Borel $\sigma$-algebra offers: a choice of all the possible intervals you might need or want.
It would make for a seriously (infinitely) long drop down menu, but it's conceptually equivalent: all the members are predefined. But you still get the convenience of choosing an arbitrary interval.
The Borel sets just function as the building blocks for the menu that is the Borel $\sigma$-algebra.
First let me clear one misconception.
The set of all subintervals is not a Borel set, but rather a collection of Borel sets. Every subinterval is a Borel set on its own accord.
To understand the Borel sets and their connection with probability one first needs to bear in mind two things:
Probability is $\sigma$-additive, namely if $\{X_i\mid i\in\mathbb N\}$ is a list of mutually exclusive events then $P(\bigcup X_i)=\sum P(X_i)$.
Therefore the collection of all events which we can measure their probability must have the property that it is closed under countable unions; trivially we require closure under complements (i.e. negation) and thus by DeMorgan we have also closure under countable intersections.
If so, the set of all events which we can measure the probability of them happening is a $\sigma$-algebra.
We wish to extend the idea that the probability that $x\in (a,b)$, where $(a,b)$ is a subinterval of $[0,1]$ is exactly $b-a$. Namely the length of the interval is the probability that we choose a point from it.
Combine these two results and we have that the Borel sets of $[0,1]$ is a collection which is a $\sigma$-algebra, and it contains all the subintervals of $[0,1]$. Since we do not want to add more than we need, then Borel sets are defined to be the smallest $\sigma$-algebra which contains all the subintervals.
Here are some very simple examples.
The set of all rational numbers in $ [0,1] $ is a Borel subset of $ [0,1] $.
More generally, any countable subset of $ [0,1] $ is a Borel subset of $ [0,1] $.
The set of all irrational numbers in $ [0,1] $ is a Borel subset of $ [0,1] $.
More generally, the complement of any Borel subset of $ [0,1] $ is a Borel subset of $ [0,1] $.
Another strange Borel set: the set of all numbers in [0,1] whose decimal expansion does not contain 7.
One more: The set of all real numbers in [0,1] whose decimal expansion contains 2 or 5. (Cantor type set)
Last but not least: The set of all real numbers in [0,1] whose decimal expansion contains only finitely many 6.
Yes, you're wrong. There's a lot more to Borel sets than intervals.
Basically any subset of $[0,1]$ that you are likely to think of is a Borel set.
The Borel $\sigma$-algebra of $[0,1]$ is the set of all Borel subsets of $[0,1]$.
Borel sets are those obtained from intervals by means of the operations allowed in a $\sigma$-algebra. So we may construct them in a (transfinite) "sequence" of steps:
$\dots$ And again and again.
And even after a sequence of steps we are not yet finished. Take, say, a countable union of a set constructed at step 1, a set constructed at step 2, and so on. This union may very well not have been constructed at any step yet. By axioms of $\sigma$-algebra, you should include it as well - if you want, as step $\infty$ (or, technically, the first infinite ordinal, if you know what that means).
And then continue in the same way until you reach the first uncountable ordinal. And only then will you finally obtain the generated $\sigma$-algebra.
This is purely based on my understanding only.
Let's start with σ-algebra: If X is a set, then all subsets of F together can be collected into the power set of X. Take any subset of the power set, say F (it is a set which is a collection of sets, or simply a collection). This obviously doesn't have all of the subsets of X. But it does have a few. If, i)X is included in the collection F, ii) all subsets in F have their complements also in F, and iii) countable unions (even intersections!) of sets in F are also available in F, then we call F a σ-algebra. What if our collection is not a σ-algebra? Say our collection of subsets of X is S. We can still add a few subsets of X to S so that we get a σ-algebra. The smallest such σ-algebra is the σ-algebra generated by S.
Now, essentially, a σ-algebra and a topology are quite different. While σ-algebra is closed under countable unions and complements, a topology is closed under arbitrary (finite or infinite) unions and finite intersections.
{Unnecessary info: So, σ-algebra imposes the additional condition of closure under complements on topology. (I am not quite sure about this!)}
An open set can be an element of a topology. A set is called "open" if, for every point inside that set, you can always find a smaller open set around that point that is completely contained within the original set. You can zoom in a bit for every point and stay entirely within the set. Open sets can come together to define a topological space satisfying all its properties.
Now, [0,1] in your example is not an open set. This is because, for the points 0 or 1 in [0, 1], you can't find another open set entirely within [0, 1] that contains 0 or 1.
The Borel σ-algebra is defined as the σ-algebra generated by the open sets. Take the set (0,1]. Consider any collection of subsets of (0,1]. Take {(0,0.5)} for example. This is a singleton collection of an open set. To generate a σ-algebra, add ∅,[0,1]. Add complement of (0,0.5), i.e., [0.5,1]. Borel σ-algebra generated by {(0,0.5)} is hence {∅, (0,1], (0,0.5), [0.5,1]}. The sets ∅, (0,1], (0,0.5) and [0.5,1] are all Borel sets. Can {[0.5,1]} generate the same? Yes! As a consequence of the condition of closure under complements, the closed set [0.5,1] also generates the same.
So your example [0,1], which is a closed set, also generates a Borel σ-algebra. Any open interval subset of [0,1] is a Borel set.
The class B of Borel sets in Euclidean space is the smallest collection of sets that includes the open and closed sets such that the countable union of sets in B are also included in B.
Correct me if I am wrong!
