\begin{definition}[Definition:Probability Distribution/General Definition]
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $\struct {S, \Sigma'}$ be a measurable space.
Let $X$ be a random variable on $\tuple {\Omega, \Sigma, \Pr}$ taking values in $\struct {S, \Sigma'}$.
Then the '''probability distribution of $X$''', written $P_X$, is the pushforward $X_* \Pr$ of $\Pr$, under $X$, on $\Sigma'$.
That is: 
{{begin-eqn}}
{{eqn	| l = \map {P_X} B
		| r = \map \Pr {X^{-1} \sqbrk B}
}}
{{eqn	| r = \map \Pr {\set {\omega \in \Omega : \map X \omega \in B} }
		| c = {{Defof|Preimage of Mapping}}
}}
{{eqn	| r = \map \Pr {X \in B}
}}
{{end-eqn}}
for each $B \in \Sigma'$, where $X^{-1} \sqbrk B$ denotes the pre-image of $B$ under $X$.
\end{definition}