What does "almost surely" mean?

Question

My lecturer has defined Brownian Motion, and one of the conditions is:

1) $W_0 =0$ almost surely

Here $(W(t))_{t\geq 0}$ is a stochastic process. What does this "almost surely" phrase mean?

Answer 1

In probability literature, "almost surely" is the equivalent of "almost everywhere" in more general measure theory.

That is, an event $A$ occurs almost surely for a probability measure $P$ iff $P(A^C)=0$.

Now, because $P$ is a probability measure (and therefore has total mass 1), this is equivalent to saying that $P(A)=1$.