ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Alphabet
Deduction system
Theory
Zermelo-Fraenkel set theory
Set
Subset
Power set
Hyperpower set sequence
Hyperpower set
Hypersubset
Subset algebra
Subset structure
Measurable space
Measurable map
Random variable
A.S. equal collection of random variables
Definition D2797
Almost sure equality relation
Remarks
Remark 0 (Almost sure equality only makes sense for random variables on the same probability space)
Note that the expression \begin{equation} \mathbb{P}(X \neq Y) = \mathbb{P} \{ \omega \in \Omega : X(\omega) \neq Y(\omega) \} \end{equation} only makes sense if $X$ and $Y$ are defined on a common probability space $P = (\Omega, \mathcal{F}, \mathbb{P})$.