Definition:Probability Measure
Definition
Let $\EE$ be an experiment.
Definition 1
Let $\EE$ be defined as a measure space $\struct {\Omega, \Sigma, \Pr}$.
Then $\Pr$ is a measure on $\EE$ such that $\map \Pr \Omega = 1$.
Definition 2
Let $\Omega$ be the sample space on $\EE$.
Let $\Sigma$ be the event space of $\EE$.
A probability measure on $\EE$ is a mapping $\Pr: \Sigma \to \R$ which fulfils the Kolmogorov axioms:
| \((1)\) | $:$ | \(\ds \forall A \in \Sigma:\) | \(\ds 0 \) | \(\ds \le \) | \(\ds \map \Pr A \le 1 \) | The probability of an event occurring is a real number between $0$ and $1$ | |||
| \((2)\) | $:$ | \(\ds \map \Pr \Omega \) | \(\ds = \) | \(\ds 1 \) | The probability of some elementary event occurring in the sample space is $1$ | ||||
| \((3)\) | $:$ | \(\ds \map \Pr {\bigcup_{i \mathop \ge 1} A_i} \) | \(\ds = \) | \(\ds \sum_{i \mathop \ge 1} \map \Pr {A_i} \) | where $\set {A_1, A_2, \ldots}$ is a countable (possibly countably infinite) set of pairwise disjoint events | ||||
| That is, the probability of any one of countably many pairwise disjoint events occurring | |||||||||
| is the sum of the probabilities of the occurrence of each of the individual events |
Definition 3
Let $\Omega$ be the sample space on $\EE$.
Let $\Sigma$ be the event space of $\EE$.
A probability measure on $\EE$ is a mapping $\Pr: \Sigma \to \R$ which fulfils the following axioms:
| \((\text I)\) | $:$ | \(\ds \forall A \in \Sigma:\) | \(\ds \map \Pr A \) | \(\ds \ge \) | \(\ds 0 \) | ||||
| \((\text {II})\) | $:$ | \(\ds \map \Pr \Omega \) | \(\ds = \) | \(\ds 1 \) | |||||
| \((\text {III})\) | $:$ | \(\ds \forall A \in \Sigma:\) | \(\ds \map \Pr A \) | \(\ds = \) | \(\ds \sum_{\bigcup \set e \mathop = A} \map \Pr {\set e} \) | where $e$ denotes the elementary events of $\EE$ |
Definition 4
Let $\Omega$ be the sample space on $\EE$.
Let $\Sigma$ be the event space of $\EE$.
A probability measure on $\EE$ is an additive function $\Pr: \Sigma \to \R$ which fulfils the following axioms:
| \((1)\) | $:$ | \(\ds \forall A, B \in \Sigma: A \cap B = \O:\) | \(\ds \map \Pr {A \cup B} \) | \(\ds = \) | \(\ds \map \Pr A + \map \Pr B \) | ||||
| \((2)\) | $:$ | \(\ds \map \Pr \Omega \) | \(\ds = \) | \(\ds 1 \) |
Also known as
In the context of probability theory, a probability measure is sometimes referred to as a probability function.
However, that name is used for something else on $\mathsf{Pr} \infty \mathsf{fWiki}$.
Also see
- Definition:Event Space: $\O \in \Sigma$ and $\Omega \in \Sigma$
- Results about probability measures can be found here.
Sources
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): probability measure