Define the Volterra operator $V$ on $C_0([0,1])\triangleq \{g \in C([0,1]):g(0)=0\}$ by $$ f \mapsto \int_0^{\cdot} f(s)ds. $$ Is there an example of an and locally positive ergodic and $V$-invariant Borel probability measure $\mu$ on $C_0([0,1])$?

Note: Locally positive means that for every non-empty open subset $U$ of $C_0([0,1])$ (with the usual compact-open topology) $\mu(U)>0$.

Define the Volterra operator $V$ on $C_0([0,1])\triangleq \{g \in C([0,1]):g(0)=0\}$ by $$ f \mapsto \int_0^{\cdot} f(s)ds. $$ Is there an example of an ergodic and $V$-invariant Borel probability measure $\mu$ on $C_0([0,1])$?

Define the Volterra operator $V$ on $C_0([0,1])\triangleq \{g \in C([0,1]):g(0)=0\}$ by $$ f \mapsto \int_0^{\cdot} f(s)ds. $$ Is there an example of an ergodic and $V$-invariant Borel probability measure on $C_0([0,1])$?

Define the Volterra operator $V$ on $C_0([0,1])\triangleq \{g \in C([0,1]):g(0)=0\}$ by $$ f \mapsto \int_0^{\cdot} f(s)ds. $$ Is there an example of an and locally positive ergodic and $V$-invariant Borel probability measure $\mu$ on $C_0([0,1])$?

Note: Locally positive means that for every non-empty open subset $U$ of $C_0([0,1])$ (with the usual compact-open topology) $\mu(U)>0$.