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 $A$.