Why is Brownian motion required to be almost surely continuous instead of merely continuous?
For example, this is stated as condition 2 in this article in section 1, Characterizations of the Wiener process. What is an example of a Brownian motion where there is a point at which the motion is not continuous?
Best Answer
You are confusing the measure on path space with Lebesgue measure. The "almost everywhere" refers to the former: almost every individual path can be taken to be continuous everywhere. Indeed, the Wikipedia page you link to says that Brownian motion is "almost surely everywhere continuous".
In other words, if $\mathbb{P}$ is Wiener measure on a suitable measurable space $(\Omega, \mathcal{F})$, then there is a set $N \subset \Omega$ of $\mathbb{P}-$measure $0$ such that for all $\omega \notin N$, $t \rightarrow \omega(t)$ is continuous for all $t \in [0,1]$.