Almost Sure Occurrence of Events and Properties

Let $(\Omega,\mathcal{F},P)$ be a probability space.

An event $A\in\mathcal{F}$ occurs almost surely (abbreviated a.s.) if

More generally, let $Q$ be a property of sample points $\omega\in\Omega$. The property $Q$ holds almost surely if there exists an event $N\in\mathcal{F}$ with $P(N)=0$ such that every $\omega\in\Omega\setminus N$ satisfies $Q$. The set of sample points satisfying $Q$ is not required to belong to $\mathcal{F}$ in this formulation; $N$ is called an exceptional event for $Q$.