On the fullness of surjective maps of an interval

Abstract:

Let $I = [0, 1]$, $\mathcal {B}$ = Lebesgue measurable subsets of $[0, 1]$, and let $\lambda$ denote the Lebesgue measure on $(I, \mathcal {B})$. Let $\tau :I \to I$ be measurable and surjective. We say $\tau$ is full, if for all $A \in \mathcal {B}$, $\lambda (A) > 0$, $\tau (A), {\tau ^2}(A), \ldots$, measurable, the condition (1) \[ \lim \limits _{n \to \infty } \lambda ({\tau ^n}(A)) = 1\] holds. We say $\tau$ is interval full if (1) holds for any interval $A \subset I$. In this note, we give an example of $\tau :I \to I$ which is continuous and interval full, but not full. We also show that for a class of transformations $\tau$ satisfying Renyi’s condition, interval fullness implies fullness. Finally, we show that fullness is not preserved under limits on the surjections.