Probability space

I had to spend lots of time understanding the probability space intuitively by relating to the real world. After understanding, I was able to spread out my thinking more than I thought before. Still, I am using this idea in my experiment and getting exciting benefits. This topic looks easy, but sorry to say that it is a really confusing topic. I will try to explain it how the way I am thinking and understanding it.

Andrey Nikolaevich Kolmogorov, a Russian mathematician, introduced the notion of probability space in the 1930s. In the modern probability theory, probability space is a very important topic in random experiments.

Let's go to Imagine. Suppose, whole space is divided into several spaces. If you measure all spaces and sum up, it should be equal to one (whole space). The scenario looks to measure space. Probability space is a measure space.

A probability space is a triple (Ω, Ƒ, P). where,

• Ω is a set of elementary outcomes
• Ƒ is a σ-algebra, i.e., Ƒ is a set with the following properties

◦ Ω ∈ Ƒ

◦ Ƒ is closed under the formation of complements

◦ Ƒ is closed under countable unions

• P is a probability measure, i.e., P is a mapping P: Ƒ→ [0, 1]

Imagine you have a house and you are standing outside of your house under a tree. Something is happening inside the house. What is happening inside the house? Whatever happening, you can call it some event. These events are happening inside your house. Here your house is Ω that included the happening events of the house. Now,

What is Ƒ? It is called σ-algebra and it is bigger than Ω. All events that can happen in the world and the happening events are included in Ƒ.

If you stand outside of your house and under the tree, you are in the world i.e. in Ƒ, not in the house (Ω). If you (event) stay in the house, you are in Ω and also in Ƒ. Why? Because your house (Ω) is located in the world (Ƒ). As I mentioned Ω is smaller than Ƒ. P is the probability measure that maps from 0 to 1. If you sum up the probability of happening of all events, it should be equal to 1.

References: 

• Billingsley, P. (1995). Probability and Measure. Wiley Series in Probability and Mathematical Statistics. Wiley, New York, 3rd ed edition.
• [Sa19] J. Sass (2019): Probability Theory. Lecture Notes. TU Kaiserslautern