Almost Sure Events

Why

We discuss negligible sets in the language of probability measures.

Definition

Suppose $(\Omega , \mathcal{A} , \mathbfsf{P} )$ is a probability space. An event $A \in \mathcal{A} $ happens almost surely (or almost certainly, almost always) if $\mathbfsf{P} (A) = 1$. (An equivalent condition is that $\mathbfsf{P} (\Omega \setminus A) = 0$.) Conversely, an event $B \subset \Omega $ happens almost never if $\mathbfsf{P} (B) = 0$.