📂Probability Theory

Probability Defined by Measure Theory

Definition ¹

Let’s say $\mathcal{F}$ is a sigma field of set $\Omega$.

1. Measurable set $E \in \mathcal{F}$ is called an Event.
2. On $\mathcal{F}$, if measure $P : \mathcal{F} \to \mathbb{R}$ satisfies $P(\Omega) = 1$, then $P$ is called Probability.
3. $( \Omega, \mathcal{F} , P )$ is called the Probability Space.

Explanation

Borrowing the strength of measure theory, we can provide a mathematical foundation for various concepts of probability theory and remove ambiguity.

1. In high school curriculum or in probability and mathematical statistics, an event was a case that could occur in any experiment. Unlike in mathematical statistics where probability was defined as a function with all events as its domain, now, elements of $\mathcal{F}$ are considered events and the term sample space is no longer used. The sigma field $\mathcal{F}$ is defined only as a formal algebraic system with the entire set $\Omega$, without worrying about exactly what kind of experiment is being conducted. Thus, there can be no ambiguity that might arise from who says what and how.
2. Probability used to be defined as a function with the sample space as its domain and $[0,1]$ as its codomain, satisfying the laws of addition of probabilities. The concept of probability redefined in measure theory does not even allow words like ‘random experiment’ or ’number of cases’. Considering the definition of measure, this definition of probability completely covers the concept of probability we have been familiar with and rigorously generalizes it.
3. The reason a new term ‘Probability Space’ is purposely defined is to regard the space $\Omega$ itself as $P$. As in basic mathematical statistics, if $\Omega = \mathbb{R}$, then $\mathcal{F}$ becomes the Borel sigma field $\mathcal{B}$, making it meaningless to discuss $(\Omega , \mathcal{F})$. It’s too simple, meaning that its applicability is limited. With the introduction of measure theory, the world of probability enters a vast phase of generalization that can be overwhelming. If you’re planning to study properly, you’ll need to be alert to how incredibly set $\Omega$ can be presented.

See Also

• Probability defined in mathematical statistics