202405240848
Status: #idea
Tags: Measure Theory
State: #nascient

Measurable Space

A sample space equipped some $\sigma -$algebra $\Xi$ is called a measurable space.
It is denoted as follows:

Probability is really just a special kind of Measure Theory.

One must understand what is a measure.
A measure is a function say $\psi$ from the $\sigma -$algebra that we elected to the interval $[0,\mathrm{\infty }]$ (this is not a typo, we can actually send values to infinity) such that:

1. $\psi (\varnothing )=0$
2. If $A_1,A_2,\dots$ is a countable collection of disjoint $\Xi -$measurable sets, then:

All we are saying is that some empty set should be measured as $0$, and that if I have some countable number of disjoint sets, I should be able to get the measure of the whole (union) by summing the individual measures.

This is eerily similar to how we define the probability of disjoint events.

The $\sigma -$algebra of Events

A Measure Space is the combination of a measurable space denoted in these notes $\Xi$ and some measure denoted $\psi$. It is written as:

We require that $\psi (\Omega )$ is well defined:

• if $\psi (\Omega )<\mathrm{\infty }$ then $\psi$ is a finite measure.
• if $\psi (\Omega )=\mathrm{\infty }$, then $\psi$ is an infinite measure
• if $\psi (\Omega )=1$, then $\psi$ is a Probability Measure (According to Kolmogorov) (yessir!)

In the last case, we denote $\psi$ with a capital $\mathbb{P}$.

So we can denote it ($\Omega ,\Xi ,\mathbb{P}$) or say that $\mathbb{P}$ is a probability measure on the measurable space ($\Omega ,\Xi$)!

So the measure axioms applied to Probabilities

1. $\mathbb{P}(\varnothing )=0$
2. $\mathbb{P}(\Omega )=1$
3. If $A_1,A_2,\dots$ are disjoint $\Xi -$measurable sets, then
   $\mathbb{P}(\bigcup _{i=1}^{\mathrm{\infty }}{A}_i)=\sum _{i=1}^{\mathrm{\infty }}\mathbb{P}(A_i)\le 1$

So as we see we can define a probability space which is nothing more than a measurable space where the measure is a probability measure space is $(\Omega ,\Xi ,\mathbb{P})$.

From now on, I'll denote the probability measure as $P$. From this, we can now define events rigorously. They are nothing more than the $\Xi -$measurable sets of our probability measure space.

This now answers why we specified "subsets under consideration" if a subset is not in our $\Xi -$algebra, then the measure defined on our measurable space does not consider it. Therefore it does not have a probability.

The Magic

There is absolutely no prescriptions (or right way) as to how to define a Probability Measure (According to Kolmogorov) or the $\sigma -$algebra for that matter.

Any arbitrary such space you can cook up, as long as you ensure it has the aforementioned properties is a totally valid probability measure. This give us a really expansible and rich soil to construct our Probability Theory.

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