Let $A$ be a generalized Volterra integral operator on $C([0,1],\mathbb{R})$ deifned by
$$
A(f)(t)\triangleq \int_0^t k(s,t)f(s) ds.
$$
Assume that $A$ is injective and let $\mu$ be a finite Borel measure on $C([0,1],\mathbb{R})$.  Then can we always find a subset $B\subseteq C([0,1],\mathbb{R})$ of arbitrarily small measure such that
$$
B \cap A^n(B)= \emptyset,
$$
for some $n \in \mathbb{N}$ depending on $\mu(C([0,1],\mathbb{R})-B)$?