To show that $\dim Z=1/2$, why do I have to show that $p\{\dim Z=1/2\}=1$?

Question

Let $(B_t)_{t\geq 0}$ a standard Brownien motion. I have to show that $\dim Z=\frac{1}{2}$ where $Z=\{t\in [0,1]\mid B_t=0\}$. Why to do this, I have to show that $$\mathbb P\{\dim Z=1/2\}=1\ \ ?$$ I don't see the correlation between $ \mathbb P\{\dim Z=1/2\}=1$ and $\dim Z=1/2$.