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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the fullness of surjective maps of an interval


Authors: Harold Proppe and Abraham Boyarsky
Journal: Trans. Amer. Math. Soc. 269 (1982), 445-452
MSC: Primary 26A18; Secondary 28D99, 58F20
MathSciNet review: 637701
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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)

$\displaystyle \mathop {\lim }\limits_{n \to \infty } \lambda ({\tau ^n}(A)) = 1$

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.

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DOI: http://dx.doi.org/10.1090/S0002-9947-1982-0637701-X
PII: S 0002-9947(1982)0637701-X
Article copyright: © Copyright 1982 American Mathematical Society