In this paper, we characterise compatibility of distributions and probability measures on a measurable space. For a set of indices \(\mathcal{J}\), we say that the tuples of probability measures \((Q_{i})_{i\in \mathcal{J}} \) and distributions \((F_{i})_{i\in \mathcal{J}} \) are compatible if there exists a random variable having distribution \(F_{i}\) under \(Q_{i}\) for each \(i\in \mathcal{J}\). We first establish an equivalent condition using conditional expectations for general (possibly uncountable) \(\mathcal{J}\). For a finite \(n\), it turns out that compatibility of \((Q_{1},\dots ,Q_{n})\) and \((F_{1},\dots ,F _{n})\) depends on the heterogeneity among \(Q_{1},\dots ,Q_{n}\) compared with that among \(F_{1},\dots ,F_{n}\). We show that under an assumption that the measurable space is rich enough, \((Q_{1},\dots ,Q_{n})\) and \((F_{1},\dots ,F_{n})\) are compatible if and only if \((Q_{1},\dots ,Q _{n})\) dominates \((F_{1},\dots ,F_{n})\) in a notion of heterogeneity order, defined via the multivariate convex order between the Radon–Nikodým derivatives of \((Q_{1},\dots ,Q_{n})\) and \((F_{1},\dots ,F_{n})\) with respect to some reference measures. We then proceed to generalise our results to stochastic processes, and conclude the paper with an application to portfolio selection problems under multiple constraints.

中文翻译：

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分配兼容性以改变度量

在本文中，我们描述了可测空间上分布和概率测度的相容性。对于一组索引\（\ mathcal {J} \），我们说概率元组测量\（（Q_ {i}）_ {i \ in \ mathcal {J}} \）和分布\（（F如果每个\（i \ ）在\（Q_ {i} \）下存在一个分布为\（F_ {i} \）在\（Q_ {i} \）下的随机变量，则{i}）_ {\ mathcal {J}}中的{i \} \）是兼容的在\ mathcal {J} \）中。我们首先使用对一般（可能不可数）\（\ mathcal {J} \）的条件期望来建立等效条件。对于有限的\（n \），结果证明\（（Q_ {1}，\ dots，Q_ {n}）\）和\（（F_ {1}，\ dots，F _ {n}）\）取决于\（Q_ {1}，\ dots，Q_ {n} \）与\（F_ {1}之间的异质性，\ dots，F_ {n} \）。我们表明，在可测量空间足够丰富的假设下，\（（Q_ {1}，\ dots，Q_ {n}）\）和\（（F_ {1}，\ dots，F_ {n}）\ ）是如果兼容且仅当\（（Q_ {1}，\点，Q _ {N}）\）占主导地位\（（F_ {1}，\点，F_ {N}）\）中的异质性的概念阶，通过\（（Q_ {1}，\ dots，Q_ {n}）\）的Radon–Nikodým导数与（（F_ {1}，\ dots，F_ {n}）的Radon–Nikodým导数之间的多元凸序定义\）关于一些参考措施。然后，我们将结果推广到随机过程，并在适用于多个约束条件下的投资组合选择问题的情况下结束本文。