On the Fullness of Surjecttve Maps of an Interval

Let / = [0, 1], ® = Lebesgue measurable subsets of [0, 1], and let A denote the Lebesgue measure on (/, ©). Let t: / —> / be measurable and surjective. We say t is full, if for all A e «, AL4) > 0, r(A), t\A), ..., measurable, the condition (1) n|im \(t"(A)) = 1 holds. We say t is interval full if (1) holds for any interval A c /. In this note, we give an example of t: / -> / which is continuous and interval full, but not full. We also show that for a class of transformations t satisfying Renyi's condition, interval fullness implies fullness. Finally, we show that fullness is not preserved under limits on the surjections.