# The Role of Sigma Algebras in Probability

I always come across these terms in probability and can never quite seem to understand what they refer to:

• Probability Spaces
• Sigma Algebras
• Filtrations

Of these terms, the term I understand the most are Probability Spaces (https://en.wikipedia.org/wiki/Probability_space). I understand that a Probability Space is characterized by 3 terms : Sample Space, Event Space, Probability Function. As a basic example, consider two dice being rolled. The Sample Space contains all 36 possible outcomes that can be obtained from two dice, the Event Space is some subset of these 36 possible outcomes (e.g. outcomes that are fully divisible by 3) and the Probability Function is a function that assigns a value between 0 and 1 to all possible events.

Sigma Algebras have been more difficult for me to understand. I consulted several references on this topic (e.g. https://www.youtube.com/watch?v=xZ69KEg7ccU) and spoke with some of my friends in Pure Mathematics who somewhat confirmed my understanding of Sigma Algebras being a "special type of subset from a set" (that meets certain criteria). I am not sure if this is correct, but in the context of Probability Spaces, it seems like the concept of a Sigma Algebra is closely related to the Event Space?

Lastly, the concept of Filtrations makes the least amount of sense to me. I also consulted several references on this topic (Example of filtration in probability theory, Confusion about probability space associated with infinite coin flips, What is meant by a filtration "contains the information" until time $t$?, Stochastic processes - Why do we need filtration?), but I am still confused about this. Here is my guess as to what a filter/filtration is:

Suppose we mix oil and water together - the oil floats above the water. In this new mixture, we can full determine where the oil is located and where the water is located. Now, suppose we mix orange juice and water together - in this new mixture, both substances "intertwine and become inseparable" from each other, and it is impossible to tell where either of these two liquids are located in the new mixture.

I think to a lesser extent, the same idea is being referred to in Probability Filters. If we have some Stochastic Process, can we attempt to determine/make inferences about the history of the Stochastic Process (e.g. which states was the Stochastic Process in?), solely based on the current state that the Stochastic Process is currently situated in? In other words, can we "filter out" this information?

• All this being said, can someone please help me understand the relationship between Probability Spaces, Sigma Algebras and Filtrations?

Thank you!

• A $\sigma$ algebra is the family of events that you can describe. Perhaps it varies during the time ..filtration means that you can decribe more and more events.. Commented Jul 1, 2022 at 16:23
• The Event Space is a sigma algebra. So it allows you to do reasonable manipulations of events: you have the universal event, the complements of events, and the (countable) unions of events. Commented Jul 1, 2022 at 16:23
• @ Thomas: thank you! I have been thinking about this the whole day! Commented Jul 2, 2022 at 0:29
• @ Henry: thank you for your clear confirmation - the event space is a sigma algebra! Commented Jul 2, 2022 at 0:30

This set of notes includes some nice interpretations:

• (pg 1) $$\sigma$$-algebra represents information. Elements of the $$\sigma$$-algebra are events whose probability are known.

• (pg 7) A filtration is a family of $$\sigma$$-algebras $$\{\mathcal{F}_t | t \geq 0\}$$ satisfying $$\mathcal{F}_s\subseteq\mathcal{F}_t$$ whenever $$s\leq t$$.

$$\mathcal{F}_t$$ can be interpreted as representing the information accumulated up to time $$t$$.

The notes give some intuition on why filtration is defined this way: in many practical applications (e.g. trading/pricing), we cannot use future information, so it is important to "respect the flow of information".

As for probability space $$(\Omega, \mathcal{F}, \mathbb{P})$$, it consists of:

• The sample space $$\Omega$$, which is the set of all possible outcomes.
• The $$\sigma$$-algebra $$\mathcal {F}$$, which is a collection of all the events we are considering.
• The probability measure $$\mathbb{P}$$, which is a set function returning an event's probability.