Certain event

An event which is known a priori to be certain to occur. More exactly, if $ \Omega = \{ \omega \} $ is the space of elementary results, an event $ A $ which occurs together with any of the elementary results $ \omega $ is said to be certain, and must clearly coincide with the entire space $ \Omega $. It is then natural to ascribe the probability 1 to a certain event:

$$ {\mathsf P} ( A) = \textrm{ measure } \{ \omega : {\omega \in A } \} = {\mathsf P} ( \Omega ) = 1 . $$

An impossible event is an event complementary to $ A $.