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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

For a continuous function $k:[0,1]^2\to \mathbb{R}$, let $A$ be the generalized Volterra integral operator on $C([0,1],\mathbb{R})$ defined by $$ A(f)(t)\triangleq \int_0^t k(s,t)f(s) ds, \quad t \in [0,1]. $$ Given a finite Borel measure $\mu$ on $C([0,1],\mathbb{R})$ does there always exist a $k$ as above such that $A$ is injective and for every $\epsilon>0$, there exists a Borel subset $B\subseteq C([0,1],\mathbb{R})$ such that

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

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 $C([0,1],\mathbb{R})$.

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 $n \in \mathbb{N}$
• $\mu(C([0,1],\mathbb{R})-B)<\epsilon$?

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

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 $\mu(C([0,1],\mathbb{R})-B)$?

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 $C([0,1],\mathbb{R})$.

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 $n \in \mathbb{N}$
• $\mu(C([0,1],\mathbb{R})-B)<\epsilon$?