📂Probability Theory

Probability Defined via Measure Theory

Definition ¹

Let $\mathcal{F}$ be a sigma field of a set $\Omega$.

1. A measurable set $E \in \mathcal{F}$ is called an event.
2. If a measure $P : \mathcal{F} \to \mathbb{R}$ on $\mathcal{F}$ satisfies $P(\Omega) = 1$, then $P$ is called a probability.
3. $( \Omega, \mathcal{F} , P )$ is called a probability space.

Explanation

Borrowing the power of measure theory provides a mathematical foundation for the various concepts of probability theory and removes ambiguity.

1. In the high school curriculum, in introductory probability, or in mathematical statistics, an event was one of the possible outcomes in a random experiment. Unlike in mathematical statistics, where probability was defined as a function whose domain is the collection of all events, now conversely we call an element of $\mathcal{F}$ an event, and the term sample space is no longer used. The sigma field $\mathcal{F}$ is defined solely in terms of the universal set $\Omega$ and a formal algebraic system on it, without caring about exactly what the random experiment is. Therefore, there can be no ambiguity that might arise depending on who says what and how.
2. Probability was a function whose domain is the sample space, whose codomain is $[0,1]$, and which satisfies the addition rule of probability. The concept of probability redefined in measure theory does not even permit words such as ‘random experiment’ or ’number of cases’. If you think about the definition of a measure, this definition of probability completely covers the previously familiar concept of probability while rigorously generalizing it.
3. The reason for going to the trouble of defining a new term ‘probability space’ is that there is now an intention to grasp the space $\Omega$ itself through $P$. As in elementary mathematical statistics, if $\Omega = \mathbb{R}$, then $\mathcal{F}$ becomes the Borel sigma field $\mathcal{B}$, and there is no point in discussing $(\Omega , \mathcal{F})$. This means it is too easy, or in other words, that the range of applicability is limited. By introducing measure theory, the world of probability enters a phase of generalization so vast as to be overwhelming. If you intend to study properly, you must brace yourself for how fantastically bizarre this $\Omega$ can be given.

See Also

• Probability as Defined in Mathematical Statistics