= Solution
For $c>0$, <Brownian scaling> and the bijection $t\mapsto c^2t$ of $[0,\infty)$ give $Z\overset d=c^{-1}Z$, or $\boxed{cZ\overset d=Z}$. The supremum is a measurable extended <random variable> because <continuity> allows the supremum over rational times, and $Z\geq0$.
For $z>0$, the function $F(z)=\mathbb P(Z\leq z)$ must satisfy $F(z)=F(z/c)$ for every $c>0$, so it is constant on $(0,\infty)$. Letting $z\downarrow0$ and $z\uparrow\infty$ shows $\mathbb P(Z=0)=\mathbb P(Z<\infty)$, hence $\boxed{Z\in\{0,+\infty\}\text{ almost surely}}$. This scaling argument alone does not decide the mass at zero; part (d) does.
Back to article page
