202405211429
Status: #idea
Tags: Continuous Probability Spaces, Probability

Borel-Sigma Algebras (or Borel-Sigma Fields)

Preamble

If we set a collection of subsets of $\Omega$ (which does not have to be a $\sigma -$algebra) we know that There Always Exist A Smallest Sigma-Algebra That Contains Any Arbitrary Collection C since we can always find the set of all $\sigma -$algebras which contain our collection $C$ and then define that smallest $\sigma -$algebra as follows. Let us define $ϝ$ as the collection of all $\sigma -$algebras which contain $C$ (this collection is never empty since $2^{\Omega }$ is one such $\sigma -$algebra) then:

Be careful to not confuse this with the idea of the $lim supϝ$ where $ϝ$ is the set of all those $\sigma -$algebra which contain our collection $C$ will be the smallest $\sigma -$algebra containing our $C$. While they look similar since $ϝ$ here is a collection of potentially uncountable sets rather than a sequence, the notion of limits does not apply here.

We know intuitively that $\sigma (C)$ is the smallest $\sigma -$algebra which contains our collection $C$ since by definition:

• It is the intersection of $ϝ$ a set defined of all $\sigma -$algebras which contain $C$, (therefore $C$ is definitely in that intersection)
• The Countable Intersection of Sigma-Algebras is Always A Sigma-Algebra

This is why we can be so brazen in defining our $\sigma (C)$ as the intersection of all those $\sigma -$algebras.

So, Borel-$\sigma$ Algebras?

Let $\Omega$ is a Topological Space, or $\Omega$ is a Metric Space or if you have no idea what those two (yet) as do I, then let $\Omega$ be some subset of ${\mathbb{R}}^n$.

All we need is the concept of Open Sets.

Then a Borel set denoted $\mathcal{B}(\Omega )$ is a $\sigma -$algebra generated by the open sets and that is defined by the space on which we are operating. In other words $\mathcal{B}(\Omega ):=\sigma (C)$ are one and the same, except that $C$ here is not an arbitrary collection but rather is specifically the set of all open sets in the topological space we are interested in. It is the smallest $\sigma -$algebra containing all open intervals.

This is amazing because the $\mathcal{B}(\Omega )$ contains just enough complexity for us to be able to do any measure of analysis without worries pun-intended, while possessing all the properties required for us to be able to use Lebesgue Measures on it. Therefore it has just the right amount of complexity to allow us to base our Probability Theory without worrying.

In fact, the concept of Borel$-\sigma -$algebras gives us a space so rich, that unless you actively attempt to construct a non-Borel set, all sets you will encounter naturally in your adventures with probability with naturally be Borel Sets. This is why we all learn how to compute integrals on Probability Density Functions which is really just a kind of Lebesgue measure without ever learning what a Borel set is.

We do lose in a sense some of the richness of our Sample Space, but as a counterpart we gain interpretability and have a space that can actually be conquered, moreover we totally side-step paradoxes like the Banach-Tarski Paradox.

Continuous Probability Spaces

This concept is of direct interest in Continuous Probability Spaces. In a similar fashion to how in Discrete Probability Spaces our defacto $\sigma -$algebra will be the power set of our sample space denoted $2^{\Omega }$, in Continuous Probability Spaces the defacto $\sigma -$algebra will be the $\mathcal{B}(\Omega$) which we often just denote $\mathcal{B}$ since its understood we are operating over the topological space represented by our sample space.

Therefore we have successfully standardized Probability Theory and can now start thinking about how to define probability measures in continuous spaces! Spoiler, we will abuse Lebesgue measures.

Relevant Links

Borel Sets
Borel Sets and Lebesgue Measure