Dependency measure for sets of random events or random variables (Q1892109)

Given a probability space \((\Omega, {\mathcal F}, P)\) and any arbitrary finite set \({\mathcal A}_ n = \{A_ 1, A_ 2, \dots, A_ n\}\) of \(n \geq 2\) events; they are said mutually independent if the following \(2^ n - n - 1\) relations hold \[ P(A_{i_ 1} A_{i_ 2} \dots A_{i_ k}) = P(A_{i_ 1})P(A_{i_ 2}) \dots P(A_{i_ k}) \tag{1} \] with \(2 \leq k \leq n\) and \(\{i_ 1, i_ 2, \dots, i_ n\} \subseteq \{1, 2, \dots, n\}\). The failure of some or all relations (1) gives rise to suitable definitions of different levels of dependency. Then an interesting dependency measure \(D_ n\), taking values in the interval \([0,1]\), is introduced, and many of its relevant properties are studied. (Reviewer's remark: The author does not assume that all the given events have probabilities different from zero and one (this would be a very strong assumption in the case of a greater than countable cardinality of the given space). Then the definition given through the product rule does not match with the intuitive meaning of independence: for example, consider the case of \(n = 2\) incompatible events, with one of them having zero probability).