On nonstandard product measure spaces
Let \((\Omega, \mathcal{A}, \mu)\) and \((T, \mathcal{T}, \nu)\) be two internal probability measure spaces. Their Loeb probability spaces are denoted by \((\Omega, L_{\mu}(\mathcal{A}), \widehat{\mu})\) and \((T, L_\nu(\mathcal{T}), \widehat{\nu})\), respectively. In this paper the authors study the relationships between the two different types of product probability spaces, namely the Loeb spaces \((\Omega \times T, L_{\mu \otimes \nu} (\mathcal{A}) \otimes (\mathcal{T}), \widehat{\mu \otimes \nu})\) and \((\Omega \times T, L_{\mu} (\mathcal{A}) \otimes L_{\mu} (\mathcal{T}), \widehat{\mu} \otimes \widehat{\nu})\) for the case that \(\widehat{\mu}\) and \(\widehat{\nu}\) are atomless. For instance, examples are presented showing the existence of a class of sets \(\{R^{s}\in L_{\mu \otimes \nu} (\mathcal{A} \otimes \mathcal{T}) : s\in[0,1] \}\) with the property that for every \(s\in[0,1], \widehat{\mu \otimes \nu} (R^{s}) = 0\) and \((\widehat{\mu} \otimes \widehat{\nu})^{\text{outer}} (R^{s}) = 1\) and for all \(s_1, s_2 \in [0,1], s_1 < s_2\) implies \(R^{s_1} \subset R^s\) and \((\widehat{\mu} \otimes \widehat{\nu})^{\text{outer}} (R^{s_2}\backslash R^{s_1}) = 1\). This shows that it may occur that there exists a continuum of increasing Loeb product null sets with large gaps with respect to the outer measure of the product of the Loeb measures. Furthermore, it is shown that for a set in the Loeb product algebra \(L_{\mu \otimes \nu} (\mathcal{A} \otimes \mathcal{T})\), its measure \(s\), its outer measure \(t\), and the inner measure \(r\) with respect to the measure \(\widehat{\mu} \otimes \widehat{\nu}\) can be completely arbitrary. The paper also contains a number of striking results involving almost independent events.
scientific article; zbMATH DE number 1839948