Consider a continuous standard brownian motion $(B_t)_{t\ge 0}$. I want to show, that $$\mathcal{L}(\{t\ge 0: B_t=0\})=0$$
I also have a hint that I need to show that $B:\mathbb{R}_+\times\Omega\longrightarrow\mathbb{R}$ is measurable. Then I shall find, that $E[\mathcal{L}(\{t\ge 0: B_t=0\})]=0$ and conclude the statemate above. I would appreciate any help! Thanks in advance!