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 $C\_0([0,1])$?



Invariant ergodic measure Volterra operator
Yes - the delta measure on the identically 0 function (and this is the only one).


