Sweep out and chaos
2005, Volume 10, Number 1, pp. 113-118
Author(s): Wang H., Xiong J.
Author(s): Wang H., Xiong J.
Let $X$ be a compact metric space and let $\mathscr{B}$ be a $\sigma$-algebra of all Borel subsets of $X$. Let $m$ be a probability outer measure on $X$ with the properties that each non-empty open set has non-zero $m$-measure and every open set is $m$-measurable. And for every subset $Y$ of $X$ there is a Borel set $B$ of $X$ such that $Y \subset B$ and $m(Y)=m(B)$. We prove that $f : (X, \mathscr{B}, m) \to (X, \mathscr{B}, m)$ sweeps out if and only if for any increasing sequence $J$ of positive integers, there is a finitely chaotic set $C$ for $f$ with respect to $J$ such that $m(C)=1$.
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