---- > [!definition] Definition. ([[probability]]) > > Suppose $\mathcal{F}$ is a [[σ-algebra]] on a set $\Omega$. > > A **probability measure** on $(\Omega, \mathcal{F})$ is a [[measure]] $P$ (often stylized $\mathbb{P}$) such that $\mathbb{P}(\Omega)=1$. > > $\Omega$ is called the **sample space**. Elements $\omega \in \Omega$ are called **elementary outcomes** or **states of the world**. > > Elements of $\mathcal{F}$ are called **events** ($\mathcal{F}$ need not be mentioned if it is clear from the context). > > If $A \in \mathcal{F}$ is an event, then $\mathbb{P}(A)$ is called the **probability** of $A$. > > The [[measure|measure space]] $(\Omega, \mathcal{F}, \mathbb{P})$ is called a **probability space**. > > ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```