Let W be a Brownian motion. The zero set of Brownian motion is the random set (1) Show that Z(ω) is a closed set for each ω. (2) Show that with probability one, every point of Z(ω) is a limit point...

Let W be a Brownian motion. The zero set of Brownian motion is the random set

(1) Show that Z(ω) is a closed set for each ω.

(2) Show that with probability one, every point of Z(ω) is a limit point of Z(ω). Conclude that Z(ω) is an uncountable set.