Hypercylic operators have very typical cyclic vectors

Question

Let $W$ be the Wiener measure on $C_0(\mathbb{R})$ and let $T\in L(C_0(\mathbb{R}),C_0(\mathbb{R}))$ be a hypercylic operator; i.e. there exists some $f \in C_0(\mathbb{R})$ such that $\{T^n(f)\}_{n=1}^{\infty}$ is dense in $C_0(\mathbb{R})$. Then, is it true that: $$ W(\left\{f\in C_0(\mathbb{R}):\, \overline{\text{span}(\{T^n(f)\}_{n=1}^{\infty})}=C_0(\mathbb{R})\right\})=1? $$