I'm reading Kolmogorov Foundations of the Theory of Probability, Page 9, which explains Independence and have questions. Kolmogorov provides the following pictured Definition I for independence. The definition is similar to what more modern books provide with exception to the inclusion of the experiment number.
The definition is confusing because an experiment $U^{(n)}$ is a decomposition of events. A decomposition is a set of disjoint sets such that when union is taken, the result is $E$. Kolmogorov typically uses $q_1, q_2, …, q_n$ typically to denote elementary events (of the sample space). From what I read, the definition implies that the intersection of an element from each experiment's decomposition.
My questions, is above explanation correct? If so, Definition I is $= 0$ because an intersection of elementary events $A^{(1)}_{q_1} \cap A^{(1)}_{q_2} \cap ….\cap A^{(n)}_{q_n} = P(0) = 0$. I think I have read something wrong, because there should be the possibility that these events intersect
Appreciate any guidance on understanding of Kolmogorov's grammatical and symbolic explanation of independence mentioned here.
Best Answer
$A_{q_i}^{(i)}$ isn't an elementary event - it's some set of events.
Each experiment $\mathfrak{A}^{(i)}$ is a decomposition of $E$ into sets $A_j^{(i)}$. Sets say $A_{q_1}^{(1)}$ and $A_{q_2}^{(2)}$ are from different decompositions, so they can intersect.