---- > [!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**. > > > > We write $\mathcal{P}\Omega$ to denote the collection of [[probability|probability measures]] on a [[σ-algebra|measurable space]] $(\Omega, \mathcal{F})$. There is extra structure at play: we have an [[covariant functor|endofunctor]] $\mathcal{P}:\mathsf{Meas}\to\mathsf{Meas}$ given by > - Sending a [[σ-algebra|measurable space]] $(\Omega, \mathcal{F})$ to the [[σ-algebra|measurable space]] $\big(\mathcal{P}(\Omega), \sigma \{ \_(A): A \in \mathcal{F} \}\big)$[^1] > - Sending a [[measurable function]] ([[random variable]]) $f:\Omega \to E$ to $\mathcal{P}X:=X_{*}(\_):\mathcal{P}\Omega \to \mathcal{P}E$.[^2] > > $\mathcal{P}$ is called the **Giry functor** on $\mathsf{Meas}$. ---- #### [^1]: By $\sigma \{ \_(A): A \in \mathcal{F} \}$ we mean the [[σ-algebra generated by a set collection|initial σ-algebra]] on $\mathcal{P}\Omega$ wrt the evaluation functions $\operatorname{eval}_{A}:\mathcal{P}\Omega \to [0,1]$ given by $\operatorname{eval}_{A}(\mathbb{P})=\mathbb{P}(A)$. [^2]: By $\__{*}(\mathbb{P})$ we mean the map $\mathbb{P}\Omega \to \mathbb{P}E$ given by $\mathbb{P} \mapsto X_{*}\mathbb{P}=\mathbb{P}_{X}$ (the [[probability distribution|law]] of $X$) where $X_{*}$ denotes [[pushforward measure|pushing forward]] along $X:\Omega \to E$. ---- #### 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 > ```