A general parameter process defined by a continuum of random variables is not jointly measurable with respect to the usual product sigma-algebra. For the case of independent random variables, a one-way Fubini extension of the product space was constructed in our 2006 paper (“Joint measurability and the one-way Fubini property for a continuum of independent random variables”, Proceedings of the American Mathematical Society, 134: 737–747) to satisfy a limited form of joint measurability. For the general case we show that this extension exists if and only if there is a countably generated sigma-algebra given which the random variables are essentially pairwise conditionally independent, while their joint conditional distribution also satisfies a suitable joint measurability condition. Applications include new characterizations of essential pairwise independence and essential pairwise exchangeability through regular conditional distributions with respect to the usual product sigma-algebra in the framework of a one-way Fubini extension.

中文翻译：

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单向 Fubini 性质和条件独立：一个等价结果

由连续的随机变量定义的一般参数过程相对于通常的乘积 sigma-algebra 是不可联合测量的。对于独立随机变量的情况，我们在 2006 年的论文中构建了乘积空间的单向 Fubini 扩展（“Joint measurability and the one-way Fubini property for a continuum of Independent random variables”，Proceedings of the American Mathematical Society , 134: 737–747) 来满足有限形式的联合可测量性。对于一般情况，我们证明这种扩展存在当且仅当存在可数生成的 sigma 代数，其中随机变量基本上是成对条件独立的，而它们的联合条件分布也满足合适的联合可测量性条件。