On Smoothing Compact Measure Spaces by Multiplication

Abstract

Let $\mu$ be a regular Borel probability measure on a compact Hausdorff space $S$, and let $m$ be Lebesgue measure on the unit interval $I$. It is proved that the measure-theoretic product $(S,\mu ) \times ({I^w},{m^w})$, where $w$ is a large enough cardinal and ${m^w}$ denotes product Lebesgue measure, is “pseudo-isometric” to $({I^w},{m^w})$. Here a pseudo-isometry $\phi$ is a point-isometry except that, instead of $\phi (A)$ being measurable for every measurable $A$, it is required only that $A$ differ by a null set from a set with measurable image. If instead $\mu$ is a Baire probability measure and $S$ is a Baire subset of ${I^w}$, then $(S,\mu ) \times ({I^w},{m^w})$ is point-isometric to $({I^w},{m^w})$. Finally it is shown that (roughly speaking) continuous maps can be “smoothed” into projection maps (to within pseudo-isometries) by multiplication by suitable projection maps.