The approximation of one-one measurable transformations by measure preserving homeomorphisms

Abstract:

This paper contains two results related to the material in [2]. Suppose $f$ is a one-one transformation of the open unit interval ${I^n}$ (where $n \geqq 2$) onto ${I^n}$. 1. If $f$ is absolutely measureable and $\varepsilon > 0$, then there is an absolutely measurable homeomorphism ${\varphi _\varepsilon }$ of ${I^n}$ onto ${I^n}$ such that $m(\{ x:f(x) \ne {\varphi _\varepsilon }(x)$ or ${f^{ - 1}}(x) \ne \varphi _\varepsilon ^{ - 1}(x)\} ) < \varepsilon$, where $m$ denotes $n$-dimensional Lebesgue measure. 2. Suppose $\mu$ is either (1) a nonatomic, finite Borel measure on ${I^n}$ such that $\mu (G) > 0$ for every nonempty open subset $G$ of ${I^n}$, or (2) the completion of a measure of type (1). If $f$ is $\mu$-measure preserving and $\varepsilon > 0$, then there is a $\mu$-measure preserving homeomorphism ${\varphi _\varepsilon }$ of ${I^n}$ onto ${I^n}$ such that $\mu (\{ x:f(x) \ne {\varphi _\varepsilon }(x)\} ) < \varepsilon$.