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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 $C_0([0,1])$?

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  • $\begingroup$ 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 $V^*$? $\endgroup$ – Jochen Glueck Nov 13 '19 at 11:54
  • $\begingroup$ A probably measure on this path space, such as the Wiener measure (if it were to satisfy these assumptions). $\endgroup$ – AIM_BLB Nov 13 '19 at 11:58
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Yes - the delta measure on the identically 0 function (and this is the only one).

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  • $\begingroup$ 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/… $\endgroup$ – AIM_BLB Nov 13 '19 at 13:05

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