# Invariant ergodic measure Volterra operator

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 $$C_0([0,1])$$?

• Do you really mean a probability measure on $C_0([0,1])$? Or do you rather mean a probability measure on $(0,1]$ which is invariant under the dual operator $V^*$? – Jochen Glueck Nov 13 '19 at 11:54
• A probably measure on this path space, such as the Wiener measure (if it were to satisfy these assumptions). – AIM_BLB Nov 13 '19 at 11:58

• If I instead consider the space $C([0,1])$ and look for locally-positive probability measures, are there other functions? Here is the formal follow-up question: mathoverflow.net/questions/345943/… – AIM_BLB Nov 13 '19 at 13:05