On nonstandard product measure spaces

Abstract

The aim of this paper is to investigate systematically the relationship between the two different types of product probability spaces based on the Loeb space construction. For any two atomless Loeb spaces, it is shown that for fixed $r \lt s$ in $[0,1]$ there exists an increasing sequence $(A_t)_{r \lt t \lt s,t\in [0,1]}$ of in the new sense product measurable sets such that $A_t$ has measure $t$ and, with respect to the usual product, the inner and outer measures are equal to $r$ and $s$, respectively. By constructing a continuum of increasing Loeb product null sets with large gaps, the Loeb product is shown to be much richer than the usual product even on null sets. General results in terms of outer and inner measures with respect to the usual product are also obtained for Loeb product measurable sets that are composed of almost independent events.