Let $A$ be a generalized Volterra integral operator on $C([0,1],\mathbb{R})$ deifned by $$ A(f)(t)\triangleq \int_0^t k(s,t)f(s) ds. $$ Fix $\mu$ be a finite Borel measure on $C([0,1],\mathbb{R})$ and suppose that we can choose $k$ such that

For every $\epsilon>0$, can we always find a Borel subset $B\subseteq C([0,1],\mathbb{R})$ such that

• $ B \cap A^n(B)= \emptyset, $ for some $n \in \mathbb{N}$
• $\mu(C([0,1],\mathbb{R})-B)<\epsilon$?
• $A$ is injective