Lebesgue measure of the set of zeroes of Brownian motion Let $B_t(w)$ be a Brownian associated with a measure $P(dw)$. Suppose that $X$ is a random set of times $t$ where $0 \le t \le 1$ such that $B_t=0$. Show that $X$ has Lebesgue measure zero $P$ almost surely.

$E(\int_Xdt) = E(\int_0^1 \chi_{B_t = 0} dt) = \int_0^1 E(\chi_{B_t = 0})dt = \int_0^1 0dt = 0$. So $\int_Xdt = 0$ almost surely
