We express the independence of real-valued random variables in terms of the conditional uncorrelation, where the conditioning takes place over the cartesian products of intervals. Next, we express the mutual independence in terms of the conditional correlation matrix. While the previous studies on the subject are based on the copula functions, our approach uses the Radon-Nikodym derivative to reduce the general problem to the simple one-dimensional conditioning.

翻译：我们利用条件不相关性来表达实值随机变量的独立性，其中条件化是在区间的笛卡尔积上进行的。接着，我们利用条件相关矩阵来表达相互独立性。尽管以往关于该主题的研究基于copula函数，我们的方法使用Radon-Nikodym导数将一般问题简化为简单的一维条件化。